fbpx

Chapter 6 – Triangles

Triangles

NCERT TEXTBOOK QUESTIONS. SOLVED
EXERCISE 6.1
1.       Fill in the blanks using the correct b given in brackets:
           (i)    All circles are ……… (congruent, similar)
           (ii)   All squares are ……… (similar, congruent)
           (iii)  All ……… triangles are similar. (isosceles, equilateral)
           (iv)  Two polygons of the same number of sides are similar, if (a) their corresponding angles are ……… and (b) their corresponding sides are ……… (equal, proportional).
Sol.   (i)    All circles are similar.
           (ii)   All squares are similar.
           (iii)  All equilateral triangles are similar.
           (iv)  Two polygons of the same number of sides are similar if:
                    (a) Their corresponding angles are equal and
                    (b) Their corresponding sides are proportional.
2.       Give two different examples of pair of.
           (i)    similar figures (ii) non-similar figures.
Sol.   (i)   (a) Any two circles are similar figures.
                   (b) Any two squares are similar figures.
           (ii)  (a) A circle and a triangle are non-similar figures.
                   (b) An isosceles triangle and a scalene triangle are non-similar figures.
3.       State whether the following quadrilaterals are similar or not:
Sol.   On observing the given figures, we find that:
           Their corresponding sides are proportional but their corresponding angles are Dot equal.
           ? The given figures are not similar.
           • Similar Triangles
           Triangles are a special type of polygons. The study of their similarity is important.
           Two triangles are said to be similar if:
           (i)  Their corresponding sides are proportional, and,
           (ii) Their corresponding angles are equal.
           THALES’ THEOREM [Basic Proportionality Theorem]
           If a line is drawn parallel to one of the sides of a triangle to intersect the other two sides in distinct points ten the other two sides are divided in the same ratio. (CBSE 2010)
           Given:
           A ?ABC in which DE || BC and DE intersects AC and AB at h and D respectively.
           To Prove:
           [ART]
                 
           Construction:
           Join BE and ‘ CD.’
           Draw EF ? AB and DC ? AC
           Proof:
           EF ? AB
           EF is height of the ?ADE, corresponding to AD.
           
           Since, ?DBE and ?ECD being on the same base DE and between the same parallel DE and BC,
           we have
           ar(?DBE) = ar(?ECD)                      …(3)
           From (1), (2) & (3), we have:
           
           Since, AD and DB are parts of AB and whereas AE and EC are parts of AC,
           ? D and E divide the sides AB and AC in the same ratio.
EXERCISE 6.2
1.       In figure (i) and (ii), DE || BC. Find EC in (i) and AD in (ii).
           Sol. (i) Since DE || BC
                         ? Using the Basic proportionality Theorem,
                         
                 
2.       E and F are points on the sides PQ and PR respectively of a A PQR. For each of the following cases, state whether EF || QR:
           (i) PE = 3.9 cm,EQ = 3 cm, PF = 3.6 cm and FR = 2.4 cm
           (ii) PE = 4 cm, QE = 4.5 cm, PF = 8 cm and RF = 9 cm
           (iii) PQ = 1.28 cm, JR = 2.56 cm, PE = 0.18 cm and PF = 0.36 cm
Sol. (i) We have: PE = 3.9 cm, EQ = 3 cm
           PF = 3.6 cm and FR = 2.4 cm
           
           
3.       In figure, if LM || CB LN || CD, prove that 
Sol. In ?ABC,
           ?LM || CB [Given]
           ?Using the Basic Proportionality Theorem, we have:
           
4.       In the figure, DE || AC and DF || AE. Prove that
5.       In the figure, DE || OQ and DF || OR. Show that EP || QR
Sol. In ?PQO
           DE || OQ                                                       [Given]
           ? Using the Basic Proportionality Theorem, we have:
           
           
           ? E and F are two distinct points on PQ and PR respetively and  E and F are dividing the two sides PQ and PR in the same ratio in ?PQR.
           ? EF || QR
6.       In the figure, A, B and C are points on OP, OQ and OR respectively such that AB || PQ and AC || PR. Show that BC || QR.
Sol. In ?PQR,
           O is a point and 0Q,- OR and joined. We have A, B and C on OP, OQ. and OR respectively such that AB || PQ and AC || OR.
           Now, in OPQ,
           AB || PQ                                                       [Given]
           
7.       Using Thales’ Theorem 6.1. prove that a line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side. (Recall that yiu have proved it in Class IX).
Sol. We Have a ?ABC in which D is the mid point of AB and E is a point on AC such that
           DE. || BC.
           DE || BC                                            [Given]
           ? Using the Basic Proportionality Theorem, we get
           
           ? E is the mid point of AC: Hence lii is proved tht “a line through the Mid-point of one side of a triangle parallel to another side bisects the third side.”
8.       Using Thales’ Theorem 6.2, prove that the line joining the mid-points of any two sides of a triangle is parallel to the third side. (Recall that you have done it in Class IX).
Sol. Given. A triangle ABE in which a line l intersects AB at D and AC at
           E, such that:
           
           To Prove:                                                       DE || BC
           Proof: (Please refer to Converse of Basic Proportionality Theorem)
9.       ABCD is a trapezium in which AB || DC and its diagonals irsect each other at the point O. Show 
Sol. We have, a trapezium ABCD such that AB || DC. The diago as AC and BD intersect each other at O.
           Let us draw OE parallel toTeither AB or DC.
           In ?ADC,
                      ?OE || DC [By construction]
           ? Using the Basic -Proportionality theorem, we get
           
10.       The diagonals of a quadrilateral ABCD intersect each other at the point 0 such that  Show that ABCD is a trapezium.
Sol. We have a trapezium ABCD in which diagonals AC end BD intersect Ich other at O such that
           
           ? Using the Basic Proportionality Theorem, we get
           
           i.e., the points 0 and E on the sides AB and AC (of ?ADB) respectively in the same ratio.
           ? Using the converse of the Basic proportionality Theorem, we have
           OE || DC and OE ||AB
           ? AB || DC
           ? ABCD is a trapezium.
EXERCISE 6.3
1.       State which pairs of triangles in the figures, are similar. Write the similarity criteria used by you for answering the question and also write the pairs of similaltfriangles in the symbolic form:
           
Sol. (i) In ?ABC and ?PQR
           We have:
                      ?A = ?P = 60°
                      ?B = ?Q = 80°
                      ?C = ?R = 40°
           ? The corresponding angles are equal,
           ? Using the AAA similarity rule,
                      ?ABC ~ ?PQR
      (ii) In ?ABC and ?QRP
           
           
2.       In the figure, ?ODC ~ ?OBA, ?BOC = 125° and ?CDO = 70°. Find ?DOC, ?DCO and ?OAB.
Sol. We have:
           ?BOC = 125° and ?CDO = 70°
           since, ?DOC + ?BOC = 180°                      [Linear Pair]
           ? ?DOC 180° – 125° = 55°                                 …(1)
           In ?DOC
           Using the angle sum property, we get
           ?DOC + ?ODC + ?DCO = 180°
           ? 55° + 70° + ?DCO = 180°
           ??DCO =180° – 55° – 70° = 55°                    …(2)
           Again,
           ?ODC ~ ?OBA [Given]
           ? Their corresponding angles are equal
           And ?OCD = ?OAB = 55°                                 …(3)
           Thus, from (1), (2) and (3)
           ?DOC = 55°, ?DCO = 55° and ?OAB = 55°.
3.       Diagonals AC and BD of a trapezium ABCD with AB || DC intersect each other at the point O. Using a similarity criterion for two triangles, show that 
Sol. We have a trapezium ABCD in which AB || DC. The diagonals AC and BD intersect at O.
           In ?OAB and ?OCD
           AB || DC                                                       [Given]
           and BD intersects them
           ??OBA = ?ODC                                      …(1) [Alternate angles]
           similarly,
           ?OAB = ?OCD                                          …(2)
           ?Using AA similarity rule,
           ?OAB ~ ?OCD
4.       In the figure,  and ?7 = ?2. Show that ?PQS ~ ?TQR.
Sol. In ?PQR
           ??1 = ?2
[Given]
           ? PR = QP
…(1) [?In a ?, sides opposite to equal angles are equal]
           
           
5.       S and T are points on sides PR and QR of ?P = ?RTS. Show that ?RPQ ~ ?RTS.
Sol. In ?PQR
           T is a point on QR and S is a point on PR such that
           ?RTS = ?P
           Now in ?RPQ and ?RTS
           ?RPQ = ?RTS
[Given]
           ?PRQ = ?TRS
[Common]
           
           ? Using AA similarity, we have
           ?RPQ and ?RTS
6.       In the figure, if ?ABE ?ACD, show that ?ADE ? ?ABC.
           
Sol. We have ?ABE ? ?ACD
           ?Their corresponding parts are equal,
           i.e., AB = AC
                  AE = AD
           
           
           ?DAE = ?BAC
[Common]
           ?Using SAS similarity, we have
           ?ADE ~ ?ABC
7.       In the figure, altitudes AD and CE of A ABC intersect each other at the point P. Show that:
           (i) ?AEP ~ ?CDP
(ii) ?ABD ~ ?CBE
(iii) ?AEP ~ ?ADB
(iv) ?PDC ~ ?BEC
Sol. We have a ?ABC in which altitude AD and CE intersect each other at P.
           ??D = ?E = 90°
…(1)
           (i) In ?EAP and ?CDP
                 ?AEP = ?CDP
[From (1)]
                 ?EPA = ?DPC
[Vertically opp. angles]
                 ?Using AA similarity, we get
                 ?EAP and ?CDP
           
           (ii) In ?ABD and ?CBE
                 ?ADB = ?CEB
[From (1)]
                 Also ?ABD = ?CBE
[Common]
                 ?Using AA similarity, we have
                 ?ABD and ?CBE
           (iii) In ?AEP and ?ADB
                 ?AEP = ?ADB
[From (1)]
                 Also ?EAP = ?DAB
[Common]
                 ?Using AA similarity, we have
                 ?AEP and ?ADB
           (iv) In ?PDC and ?BEC
                 ?PDC = ?BEC
[From (1)]
                 And ?DCP = ?ECB
[Common]
                 ?Using AA similarity, we get
                 ?PDC and ?BEC
8.       E is a point on the side AD produced of a parallelogram.ABCD and BE intersects CD at F. Show that ?ABE ~ ?CFB.
Sol. We,have a parallelogram ABCD in which AD is produced to E and BE is joined such that BE intersects CD at F.
           Now, in ?ABE and ?CFB
                 ?BAE = ?FCB
[Opp. angles of a || gm are always equal]
                 ?AEB = ?CBF
[?Parallel sides are intersected by the transversal BE]
           Now, using AA similarity, we have
                 ?ABE and ?CFB
9.       In the figure, ABC and AMP are two right triangles, right angle at B and M respectively. Prove that:
           
Sol. We have ?ABC, right angled at B and ?AMP, right angled at M.
                 ??B = ?M = 90°
…(1)
           (i) In ?ABC and ?AMP
                 ?ABC = ?AMP
[From (1)]
                 And ?BAC = ?MAP
[Common]
                 ? Using AA similarity, we have
                 ?ABC ~ ?AMP
           (ii) ?ABC ~ ?AMP
[As proved above]
                 ? Their corresponding sides are proportional.
                 
10.       CD and GH are respectively the bisectors of ?ACB and ?EGFsuch that D and H lie on sides AB and FE of ?ABG amd ?EFG respectively. If ?ABC ~ ?FEG, show that:
           
Sol. We have two similar ?ABC and ?FEG such that CD and GH are the bisectors of ?ACB and ?FGE respectively.
           (i) In ?ACD and ?FGH
                 ?CAD = ?GFH
…(1)[?ABC ~ ?FEG ??A =?F]
                 since ?ABC ~ ?FEG
[Given]
                 ??C = ?G
                 ??ACD ~ ?FGH
…(2)
                 From (1) and (2),
                 ?ACD ~ ?FGH
                 ? Their corresponding sides are proportional,
                 
           (ii) In ?DCB and ?HGE
                 ?DBC = ?HEG
…(1)[?ABC ~ ?FEG ??B = E]
                 Again, ?ABC ~ ?FEG ??ACB ~ ?FGE
                 
                 ??DBC = ?HEG
…(2)
                 From (1) and (2), we get
                 ?DCB ~ ?HGE
[AA similarity]
           (iii) In ?DCA and ?HGF
                 ?DAC = ?HFG
…(1) [?ABC ~ ?FEG ??CAB = ?GFE ? ?CAD = ?GFF ??DAC = ?HFG]
                 Also ?ABC ~ ?FEG ??ACB ~ ?FGE
                 
11.       In the figure, E is a point on side CB produced of an isosceletriangle ABC with AB = AC. If AD ? BC an EF ? AC, prove that ?ABD ~ ?ECF,
Sol. We have an isosceles ?ABC in which AB = AC.
           In ?ABD and ?ECF
           AB = AC
[Given]
           ?Angles opposite to them are equal
           ??ACB = ?ABC
           ??ECF = ?ABD
…(1)
           Again AD ? BC and EF ? AC
           ??ADB = ?EFC = 90°
…(2)
           
           From (1) and (2), we have
           ?ABD ~ ?ECF
[AA criteria of similarity]
12.       Sides AB and BC’a d median AD of.hqriangle ABC are respectively proportional to sides PQ and QR and median PM of ?PQR (see figure). Show that ?ABC ~ ?PQR.
Sol. We have ?ABC and ?PQR in which AD and PM are medial sfeorfesponding to sides BC and QR respectively such, that
           
           ?Using SSS similarity, we have:
           Their corresponding q es are equal
           ??ABD = ?PQM
           ??ABC = ?PQR
           Now, in ?ABC and ?PQR
           
13.       D is a point on the side BC of a triangle ABC such that ?ADC = ?BAC. Show that CA2 = CB . CD.
Sol. We have a ?ABC and point D on its side BC such that
           ?ADC = ?BAC
           In ?ABC and ?ADC
                 ?BAC = ?ADC
[Given]
           And ?BCA = ?DCA
[Common]
           
           ? Using AA similarity, we have
           ?BAC ~ ?ADC
           ?Their corresponding sides are proportional
           
14.       Sides AB and AC and median AD of a triangle ABC are respectively proportional to sides PQ and PR and median PM of another ,triangle PQR. Show that ?ABC ~ ?PQR.
Sol. We have two As ABC and PQR such that AD and PM are medians corresponding to BC and QR respectively. Also
           
           From (1), we have:
           
           ??ABD ~ ?PQM.
[using SSS similarity]
           since, the corresponding angles of similar triangles are equal.
           ??ABD = ?PQM
           ? ?ABC = ?PQR
…(2)
           Now, in ?ABC ~ ?PQR
           ?ABC = ?PQR
[From (2)]
           
15.       A vertical pole of length 6 m casts a shadow 4 m long on the ground and at the same time a tower casts a shadow 28 m long. Find the height of the tower.
Sol. Let AB = 6 m be the pole and BC = 4 m be its shadow (in right ?ABC), whereas DE and EF denote the tower and its shadow respectively.
           EF = Length of the shadow of the tower = 28 m
           And DE = h = Height of the tower
           In ?ABC and ?DEF we have
           ?B = ?E = 90°
           ?A = ?D     [Angular elevation of the sun at the same time.]
           
           ?Using AA criteria of similarity, we have
           ?ABC ~ ?DEF
           ?Their sides are proportional
           
           Thus, the required height of the tower is 42 m.
16.       If AD and PM are medians of triangles ABC and PQR, respectively where ?ABC ~ ?PQR, prove that 
Sol. We have ?ABC ~ ?PQR such that AD and PM are the medians corresponding to the sides BC and QR respectively.
           
           ??ABC ~ ?PQR
           And the corresponding sides of similar triangles are proportional.
           
           ?Corresponding angles are also equal in two similar triangles
           ??A = ?P, ?B = ?Q and ?C = ?R                                            …(2)
           Since AD and PM are medians
           ?BC = 2 BD and QR = 2 QM
           ?From (1),
           
EXERCISE 6.4
1.       Let ?ABC ~ ?DEF and their areas be, respectively, 64 cm2 and 121 cm2. If EF = 15.4 cm, find BC.
Sol. We have
           ar (?ABC) = 64 cm2
           ar (?DEF) = 121 cm2 and EF = 15.4 cm
           ?ABC ~ ?DEF                                            [Given]
           
2.       Diagonals of a trapezium ABCD With AB || DC intersect each other at the point O. If AB = 2 CD, find the ratieof the areas of triangles AOB and COD.
Sol. We have in trap. ABCD, AB || DC.
           Diagonals AC and BD intersect at O.
           In ?AOB and ?COD
           ?AOB = ?COD,                                            [Vertically opposite angles]
           ?OAB = ?OCD,                                            [Alternate angles]
           ?Using AA criterion of similarity, we have:
           ?AOB ~ ?COD
           
3.       In the figure, ABC and DBC are two triangles on the same base BC. If AD intersects BC at O, show that 
Sol. We have:
           ?ABC and ?DBC are on the same base BC. Also BC and
           AD intersect at O.
           Let us draw AE ? BC and DF ? BC.
           In ?AOE, ?AEO = 90° and
           In ?DOF, ?DFO = 90°
           ??AEO = ?DFO                                 …(1)
           Also, ?AOE = ?DOF                      …(2) [Vertically Opposite Angles]
           ? From (1) and (2),
           ?AOE ~ ?DOF                                 [By AA similarity]
           ? Their corresponding sides are proportional
           
           
4.       If the areas of two similar triangles are equal, prove that they dre congruent.
Sol. We have ?ABC and ?DEF, such that ?ABC ~ ?DEF and ar(?ABC) = ar (?DEF).
           Since, the ratio of areas of two sinular triangles is equal to the square of, the ratio of their corresponding sides.
           
5.       D, E and F are respectively the mid-points of sides AB, BC and CA of ?ABC. Find the ratio of the areas of ?DEF and ?ABC.
Sol. We have a ?ABC in which D, E and F are mid points of AB, AC and BC respectively. D, E and F are joined to form ?DEF.
           Now, D is mid-point of AB
           
           ? Using the converse of the Basic Proportionality Theorem, we have
DE || BC
           ??ADE = ?ABC                                 …(3) [Corresponding angles]
           Also ?AED = ?ACE                                 …(4) [Corresponding angles]
           Now from (3) and (4), we have
           ?ABC ~ ?DEF                                 [Using AA similarity]
           
6.       Prove that the ratio of the areas of two similar triangle is equal to the square of the ratio of their corresponding medians.
Sol. We have two triangles ABC and DEF such that
           ?ABC ~ ?DEF
           AM and DN are medians corresponding to BC and EF respectively.
           ?ABC ~ ?DEF
           ? The ratio of their areas is equal to the square of the ratio of their corresponding sides.

]

           
7.       Prove that the area of an equilateral triangle described on one side of a square is equal to half the area o the equilateral triangle described on one of its diagonals.
Sol. We have a square ABCD, whose diagonal AC. Equilateral ?BQC is described on the side BC and another equilateral ?APC is described on the diagonal AC.
           All equilateral triangles are similar.
           ??APC ~ ?BQC
           ?The ratio of they areas is equal to the square of the ratio of their corresponding sides.
           
8.       ABC and BDE are two equilateral triangles such that D is the mid-point of BC. Ratio of the areas of triangles ABC and BDE is
           (A) 2 : 1
(B) 1 : 2
(C) 4 : 1
(D) 1 : 4
Sol. We have an equilateral ?ABC and D is the mirpoint of BC. DE is drawn such that BDE is also an equilateral triangle.
           Since, all equilateral triangles-are sirhilar,
           ??ABC ~ ?BDE
           ?The ratio of their areas is, equal to the square of the ratio of their corresponding sides.
           
           From (1) and (2), we have:
           
9.       Sides of two similar triangles are in the ratio 4: 9. Areas of these triangles are in the ratio
           (A) 2 :13
(B) 4 : 9
(C) 81 : 16
(D) 16 : 81
Sol. We have two similar triangles such that the ratio of their corresponding sides is 4 : 9
           ? The ratio of areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
           
EXERCISE 6.5
1.       Sides of triangles are given below. Determine which of them are right triangle. In case of a right triangle, write the length of its hypotenuse.
           (i) 7 cm, 24 cm, 25 cm
           (ii) 3 cm, 8 cm; 6 cm
           (iii) 50 cm, 80 cm, 100 cm
           (iv) 13 cm, 12 cm, 5 cm
Sol. (i) The sides are:
           Here, (7 cm)2 = 49 cm2
           (24 cm)2 = 576 cm2
           (25 cm)2 = 625 cm2
           (49 + 576) cm2 = 625 cm2
           The given ? is a right triangle. Hypotenuse = 25 cm.
           (ii) The sides, are:
           Here, (3 cm)2 = 9 cm2
           (8 cm)2 = 64 cm2
           (6 cm)2 = 36 cm2
           (9 + 36) ? 64 cm
           It is not a right triangle.
           (iii) The sides are:
           50 cm, 80 cm, 100 cm
           Here, (50 cm)2 = 2500 cm2
           (80 cm)2 = 6400 cm2
           (100 cm)2 = 10000 cm2
           (2500 + 6400) cm2 ? 10000 cm2
           ? It is not a right triangle.
           (iv) The sides are:
           13 cm, 12 cm, 5cm
           Here, (13 cm)2 = 169 cm2
           (12 cm)2 = 144 cm2
           (5 cm)2 = 25 cm2
           (144 + 25) cm2 = 169 cm2
           ? The given triangle is a right triangle.
2.       PQR is a triangle, right angled at P and M is a point on QR such that PM ? QR. Show that PM2 = QM MR.
Sol. In ?QMP and ?QPR,
           ?QMP = ?QPR                      [Each = 90°]
           ?Q = LQ                      [Common]
           ??QMP ~ ?QPR                      …(1) [AA similarity]
           Again in ?PMR and ?QPR,
           ?PMR = ?QPR                                 [Each = 90°]
           ?R = ?R                                            [Common]
           ??RMP ~ ?QPR                                 …(2) [AA similarity]
           
3.       In the figure, ABD is a triangle, right angled at A and AC ? BD. Show that
           (i) AB2 = BC . BD
(ii) AC2 = BC. DC
(iii) AD2 = BD CD
Sol. (i) In ?BAC and ?BDA
           ?ACB = ?BAD                      [Each = 90°]
           ?B = ?B                                 [Common]
           ?BAC ~ ?BDA                      [AA similarity]
           
           ?AB2 = BC . BD
           (ii) In ?ACB and ?DCA
           ?ACB = ?DCA                      [Each = 90°]
           ?C = ?C                                 [Common]
           ??ACB ~ ?DCA                      [AA similarity]
           
           ? Their corresponding sides are proportional
           
4.       ABC is an isosceles triangle, right angled at C. Prove that AB2 = 2AC2.
Sol. We have right ?ABC such that?C = 90° and AC =.BC.
           
           ?By Pythagoras Theorem, we ave
           AB2 = AC2 + BC2
           = AC2 + AC2                                 [AB = AC (given)]
           = 2 AC2
           Thus, AB2 = 2AC2
5.       ABC is a right angled triangle with AC = BC. If AB2 = 2AC, prove that ABC is a right triangle.
Sol. We have a ?ABC such that AB = AC.
           
           Also, AB2, 2 AC2
           ? AB2 = AC2 + AC2
           But AC = BC
           But AB2 = AC2 + BC2
           ? Using the converse Pythagoras Theorem,
6.       ABC is an equilateral triangle of side 2a. Find each of its altitudes.
Sol. We have an equilateral ?ABC in which AB = AC = CA = 2a.
           Let us draw CD ? AB i.e., CD is an altitude corresponding to AB.
           Now, in ?ACD and ?BCD,
           AC = BC                                                       [Each = 2a]
           CD = CD                                                      [Common]
           ?ADC = ?BDC                                        [Each = 90°]
           ?ACD ? ?BCD                                       [By AA congruency]
           
           ? Their corresponding parts are equal.
           ? AD = BD
           i.e., D is the mid point of AB
           
           Now in right ?ADC, we have
           AC2 = AD2 + CD2
           ?CD2 = AC2 – AD2
           = (2a)2 – (a)2
           = 4a2 – a2 = 3a2
           
           Similarly, each of the other altitudes are  [ Each side of an equilateral ? is equal]
7.       Prove that the sum of the squares of the side of a rhombus is equal to the sum of the squares of its diagonals.
Sol. Let us have a rhombus ABCD.
           Diagonal of a rhombus bisect each otherat right angles.
           ? OA = OC and OB and OD
           Also, ?ADB = ?BOC                                 [Each = 90°]
           And ?COD = ?DOA                                 [Each = 90°]
           In right, ?AOB,
           We have
           AB2 = OA2 + OB2                                 …(1) [Using Pythagoras theorem]
           
           Similarly,
           BC2 = OB2 + OC2                                 …(2)
           CD2 = OC2 + OD2                                 …(3)
           and DA2 = OD2 + OA2                                 …(4)
           Adding (1), (2), (3) and (4),
           AB2 + BC2 + CD2 + DA2
           = [OA2 + OB2] + [OB2 + OC2]+ [OC2 + OD2] + [OD2 + OA2]
           = 2OA2 + 2OB2 + 2OC2 + 2OD2
           = 2 [OA2 + OB2 + OC2 + OD2]
           = 2[OA2 + OB2 + OA2 + OB2]
                                 [OA = QC and OB = OD]
           = 2 [2 OA2 + 2 OB2]
           
           
           Thus, sum of the squares of the sides of a rhombus is equal to theuq of thg,squares of its diagonals.
8.       In the figure, O is a point in the interior of a triangle ABC, OD ? BC, OE ? AC and OF ? AB. Show that
           (i) OA2 + OB2 + OC2 – OD2 – OE2 – OF2 = AF2 + BD2 + CE2,
           (ii) AF2 + BD2 + CE2 = AE2 + CD2 + BF2.
           
Sol. We have a pointlin the interior of a ?ABCsulhat QD ? BC, OE ? AC and On ? AB.
           (i) Let us join OA, OB and OC.
           In right ?OAF, we have
           OA2 = OF2+ AF2                                            [Using Pythagoras Theorem]
           Similarly, from right triangles ODB and OEC, we have
           OB2 = BD2 + OD2, and
           OC2 = CE2 + OE2
           
           Adding,
           OA2 + OB2 + OC2 = (AF2 + OF2) + (BD2 + OD2) + (CE2 + OE2)
           ? OA2 + OB2 + OC2 = AF2 + BD2 + CE2 + (OF2 + OD2 + OE2)
           ?OA2 + OB2 + OC2 (OD2 + OE2 + OF2) = AF2 + BD2 + CE2
           ? OA2 + OB2 + OC2 – OD2 – OE2 – OF2 = AF2 +BD2 + CE2
           (ii) In right triangles OBD and OCD:
           OB2 = OD2 + BD2                                 [Using Pythagoras Theorem]
           and OC2 =OD2 + CD2
           ? OB2 – OC2 = OD2 + BD2 – OD2 – CD2
           ? OB2 – OC2 = BD2 – CD2
           Similarly, we have
           OC2 – OA2 = CE2 – AE2                                 …(2)
           and OA2 – OB2 = AF+ – BF2                                 …(3)
           Adding (1), (2) and (3) we get:
           (OB2 – OC2) + (OC2 – OA2) + (OA2 – OB2) = (BD2 – CD2) + (CE2 – AE2) + (AF2 – BF2)
           ? 0 = BD + CE2 + AF2 – (CD2 + AE2 – BF2)
           ? BD2 + CE2 + AF2 = CD2 + AE2 + BF2
           or AF2 + BD2 + CE2 = AE2 + BF2 + CD2
9.       A ladder 10 m long reaches a window 8 m above rah el ouundd.. Find the distance of the foot of the ladder from base of the wall.
Sol. Let PQ be the ladder = PQ = 10 m
           PR, the wall ?PR = 8 m
           RQ is the base = RQ = ?
           Now, in the right ?PQR,
           PQ2 = PR2 + QR2
           ? 102 = 82 + QR2
           [using Pythagoras theorem]
           
           ? QR2 = 102 – 82
           = (10 +8) (10 – 8)
           = 18 × 2 = 36
           
           Thus, the distance of the foot of the ladder from’ the base to the all is 6 m.
10.       A guy wire attached to a vertical pole of height 18 m is 24 m long and has a stake attached to the other end. How far from the base of the should the stake be driven so that the wire will be taut?
Sol. Let AB is the wireand BC is the vertical pole. The point A is the stake.
                                 ? AB = 24 m, BC = 18 m
           Now, in the right ?AAC, using Pythagoilas Theorem, we have:
           
           AB2 = AC2 + BC2
           242 = AC2 + 182
           AC2 = 242 – 182
           = (24 – 18) (24 + 18)
           = 6 × 42 = 252
           = 7 × 36
           
           Thus, the stake is required to be taken at  from the base of the pole to make the wire taut.
11.       An aeroplane leaves an airport and flies due pgrth a speed of 1000 km per hour. At thejsame time, another aeroplane leaves the same airport a djliesti due west at a sped of 1200 km per hour. flow far apart will be the two planes after  hours?
Sol. Let the point A represent the airport.
           Let the plane-I fly towards North
           
           ? Distance of the plane-l from the airport after  hours
           = speed × time
           
           Let the plane-II flies towards West,
           
           ? Distance of the plane-lI from the airport after  hours
             
             = 1800 km
           Now, in right ?ABC, using Pythagoras theorem, we have:
             BC2 = AB2 + AC2
             ?BC2 = (1500)2 + (1800)2
             = 2250000 + 3240000
             = 5490000
             
             Thus, after  hours the two planes are  apart from each other.
12.       Two poles.o heighfs 6 m and 11 m stand on a plane ground. If the distance between the feet of the poles is 12 m, find the distancebetwe ten tops.
Sol. Let the two poles AB and CD are such that the distance between their feet AC = 12 m.
             ? Height of poleI AB = 11
             
             Height of pole-II, CD = 6 m
             ? Extra height of pole-I = BE = 11m – 6m = 15m
             Let us join the tops of the poles D and B.
             Now, in rt. ?BED, using the Pythagoras Theorem, we have:
             DB2 = DE2 + EB2
             ? DB2 = 122 + 52
             = 144 + 25 = 169
             
             Thus, the required distance between the tops =13 m.
13.       Dad E are points on the sides CA and CB respectively of a triangle ABC, right angled at C. Prove that AE2 + BD2 = AB2 + DE2.
Sol. We have a right ?ABC such that ?C = 90°. Also D and E are points on CA and CB respcetively. Let us join AE and BD.
           In right ?ACB,
           Using Pythagoras Theorem, we have
           AB2 = AC2 + BC2                                 …(1)
           Using Pythagoras Theorem,
           DE2 = CD2 + CE2                                 …(2)
             
             Adding (1) and (2), we get
             AB2 + DE2 = [AC2 + BC2] + [CD2 + CE2]
             = AC2 + BC2 + CD2 + CE2
             = [AC2 + CE2] + [BC2 + CD2]
             From the figure, we can have
             In right ?ACE,
             [AC2 + CE2] = AE2 and
             In right ?BCD,
             [BC2 + CD2] = BD2
             AB2 + DE2 = AE2 + BD2
             or AE2 + BD2 = AB2 + DE2
14.       The perpendicular from A on side BC of a ?ABC intersects BC at D such that DB = 3 CD (see figure). Prove that 2AB = 2AC2 + BC2.
Sol. We have, a ?ABC such that AD ? BC. The position of D is such that, BD = 3 CD.
             In right ?ABC, we have                                       [using Pythagoras Theorem] …(1)
             AB2 = AD2 + BD2
             Similarly from right ?ACD, we have:
             AC2 = AD2 + CD2                                       …(2)
             Subtracting (2) from (1), we get
             AB2 – AC2 = DB2 – CD2                                       …(3)
             Now BC = DB = CD
             = 3CD + CD = 4 CD [BD = 3 CD]
             
             
             Now, substituting the values of CD and BD in (3), we get
             
15.       In an equilateral triangle ABC, D is a point on side BC such that  Prope that 9 AD2 = 7 AB2.
Sol. We have an equalateral A ABC; in which D is a point on BC such that 
             Let us draw 
             In right ?APB, we have
             AD2 = AP2 + BP2                                       …(1) [using Pythagoras Theorem]
             In right ?APD, we have
             AD2 = AP2 + DP2                                       …(1) [using Pythagoras Theorem]
             AP2 = AD2 – DP2
             ? From (1), we have
             AB2 = (AD2 – DP2) + BP2
             
             
16.       In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one of its altitudes.
Sol. We hae an equilateral ?ABC, in which AD ? BC.
             Since, an altitude in an equilateral ?, that bisects the corresponding side.
             ?D is the mid point of BC.
             
             ? 4 AB2 = 4 AD2 + BC2
             ? 4 AB2 = 4 AD2 + AB2
             ? 4 AD2 = 4 AB2 – AB2
             ? 4 AD2 = 3 AB2
             ? 3 AB2 = 4 AD2
17.       Tick the correct answer and justify: In ?ABC,, AB =  cm, AC = 12 cm that BC = 6 cm. The angle B is:
             (A) 120°
(B) 60°
(C) 90°
(D) 45°
Sol. We have AB =  cm, AC = 12 cm and BC = 6 cm
           ?AB2 = ()2 = 36 × 3 = 108
           AC2 = 122 = 144
           BC2 = 62 = 36
           Since 144 = 108 + 36
           i.e., AC2 = AB2 = BC2
           ? The given sides form a triangle ?ABC right angled at B.
           ? B = 90°
           ? The correct answer is (C) 90°.
EXERCISE 6.6 (OPTIONAL)
Q1.       It: the figure, PS is the bisector of ?QPR of d PQR. Prove that 
Sol.   We have ?PQR in which PS is the bisector of ?QPR
                  ?QPS = ?RPS
                  Let us draw RT || PS to meet QP produced at T, such that
                  ?1 = ?RPS                                                      [Alternate angles]
            Also ?3 = ?QPS                                                 [Corresponding angles]
            But ?RPS = ?QPS                                             [Given]
                  ?1 = ?3
                  ? PT = PR
[Equal sides of a triangle opposite to equal angles]
            Now, in ?QRT,
                  ?PR || RT                                                      [By construction]
            Using the Basic Proportionality Theorem, we have:
                  
Q2.       In the figure, ? is a point on hypotenuse AC of ?ABC, such that
            BD ? AC, DM ? BC and DN ? AB. Prove that.
            (i)   DM2 = DN.MC            (ii)   DN2 = DM AN
Sol.   We have AC as the hypotenuse of ?ABC.
            Also BD ? AC, DM ? BC and DN ? AB
            ? BMDN is a rectangle.
            ?BM = ND                                                               [Opp. sides of a rectangle]
            (i) In ?BMD and DMC,
                  ?DMB = 90° = ?DMC                                 …(1)
                  ?3D ? AC                                                        [Given]
                  ?1 + ?2 = 90°
                  In ?BDM,
                  ?3 + ?2 = 90°
                  ??1 = ?3                                                          …(2)
                  From (1) and (2)
                  ??BMD ~ ?DMC                                     [By AA similarity]
                  ? Their corresponding sides are proportional.
                  
                  [? DN and BM are opposite sides of a rectangle, ? DN= BM ]
                  ? DN×MC = DM×DM
                  ? DN×MC = DM2
                  or DM2 = DN × MC
            (ii) In ?BND and ?DNA, we have:
                  ?BND = ?DNA                                    [Each = 90°]
                  ?DBN = ?ADN                                    [As in part (i)]
                  ??BND ~ ?DNA [AA similarity]
                  ? Their corresponding sides are proportional.
                  
Q3.       IN the figure, ABC is a triangle in which ?ABC > 90° and AD ? CB produced. Prove that
                  AC2 = AB2 + BC2 + 2BC.BD.
Sol.      ABC is a right triangle ?ABC > 90° and AD ? CB
            In ?ADB,
                  ?D = 90°
            ? Using pythagoras Theorem, we have:
            AB2 = AD2 + DB2                            …(1)
            Again in ?ADC,
                  ?D = 90°
            ? Using Pythagoras theorem,
                  AC2 = AD2 + DC2
                  =AD2 + [BD + BC]2
                  = AD2 + [BD2 + BC2 + 2BD BC]
                  ? AC2 = [AD2 + DB2] + BC2 + 2BC BD
                  ? AC2 = AB2 + BC2 + 2BC BD                                          [From (1)]
                  Thus, we have:
AC2 = AB2 + BC2 + 2BC BD
Q4.       In the figure, ABC is a triangle in which ?ABC < 90° and AD ? BC. Prove that
                  AC2 = AB2 + BC2 – 2BC.BD.
Sol.   We have ?ABC in which ?ABC < 90° and AD ? BC.
            In right ?ADB,
                  ?D = 90°
            ? Using Pythagoras theorem, we have:
                  AB2 = AD2 + BD2
            Also in right ?ADC,
                  ?D = 90°
            ? Using Pythagoras Theorem, we have:
                  AC2 = AD2 + DC2
                  = AD2 + [BC . BD]2
                  = AD2 + [BC2 + BD2 – 2BC.BD]
                  = [AD2 + BD2] + BC2 – 2BC.BD
                  = [AB2] + BC2 – 2BC.BD,                                    From (1)]
            Thus, AC2 = AB2 + BC2 – 2 BC.BD
                  which is the required relation.
Q5.       In the figure, AD is a median of a triangle ABC and AM ? BC. Prove that.
            
Sol.   We have ?ABC in which AD is median and AM ? BC such that
                  ?ADC > 90° and ?ADM > 90°
            (i) IN ?AMC,
                  ?M = 90°
                  AC2 = AM2 + MC2
                  = AM2 + MD2 + DC2 + 2MD.DC
                  = AD2 + DC2 + 2MD. DC [MD2 + AM2 = AD2]
            
            (ii) In ?AMB,
                  ?AMC = 90°
                  ? Using Pythagoras theorem, we have:
                  AB2 = AM2 + BM2
                  = AM2 + (BD – DM)2
                  = AM2 + BD2 + DM2 + 2BD.DM
                  = AD2 + BDC2 – 2BD. DM                                                      [DM2 + AM2 = AD2]
                  
            (iii) Adding (1) and (2) we get,
                  
Q6.       Prove that the sum of the squares of the diagonals of parallelogram is equal to the sum of the squares of its sides.
Sol.   We have a parallelogram ABCD.
            AC and BD are the diagonals of ||gm ABCD.
            Diagonals of a || gm bisect each other.
            ? O is the mid-point of AC and BD.
            Now, in ?ABC,
            BO is a median
                  
                  Also, in ?ADC,
                  Do is a median,
                  
Q7.       In the figure, two chords AB and CD intersect each other at the point P. Prove that.
               (i)   ?APC ~ ?DPB                              (ii)   AP.PB = CP. DP
Sol.   We have two chords, AB and CD of a circle. AB and CD intersect at P.
            ??AOC = ?DPB                                    [Vertically opp. angles] …(1)
            Let us join AC and BD.
            (i) In ?APC and ?DPB,
                  ?CDP = ?BDP                                    [Angles in the same segment]…(2)
            From (1) and (2) and using AA similarity we have
                  ?APC ~ ?DPB
            (ii) Since ?APC ! ?DPB                                                [As proved above]
                  ? Their corresponding sides are proportional,
                  
Q8.       In the figure, two chords AB and CD of a circle intersect each other at the point P (when produced) outside the circle. Prove that.
               (i) ?PAC ~ ?PDB                              (ii) PA.PB =PC.PD
Sol.   We have two chords AB and CD, when produced meet outside the circle at P.
            (i) Since, in cyclic quadrilateral, the exterior angle is equal to the interiror opposite angle,
                  ?PAC = ?PDB                                            …(1)
                  and ?PCA = ?PBD                                    …(2)
                  ? From (1) and (2) and using the AA similarity, we have
                  ?PAC ~ ?PDB
            (ii) Since, ?PAC ~ ?PDB
                  ? Their corresponding sides are proportional.
                  
Q9.       In the figure, D is a point on side BC of ?ABC such that  Prove that AD is the bisector of ?BAC.
Sol.   Let us produce BA to E such that
                  AE = AC
            Join EC.
            
            And BE is a transversal,
                  ?BAD = ?AEC                                                 [Corresponding angles] …(1)
            Also ?CAD = ?ACE                                            [Alternate angles] …(2)
            Since, AC = AE
            ? Their opposite sides are equal
            ??AEC = ?ACE                                    …(3)
            From (1) and (3), we have
                ?BAD = ?ACE                                    …(4)
            From (2) and (4), we have
                ?BAD = ?CAD
            ? AD is bisector of ?BAC.
Q10.       Nazima is fly fishing in a stream. The tip of her fishing rod is 1.8m above the surface of the water and the fly at the end of the string rests on the water 3.6 m away and 2.4m from a point directly under the tip of the rod. Assumring that her string (from the tip of hr rod to the fly) is taut, how much string does she have out (see figure)? is she pulls in the string at the rate of 5 cm per second, what will be the horizontal distance of the fly from her after 12 secdons?
Sol.   Let us find the lenght of the string that she has out.
            In right ?ABC,
                AC2 = AB2 + CB2                                [using Pythagoras Theorem.]
                AC2 = (2.4)2 + (1.8)2
                ? AC2 = 5.76 + 3.24 = 9.00
                
                i.e. Length of string she has out = 3m
                Since, the string is pulled out at the rate of 5 cm/ sec,
                Length of the string pulled out in 12 seconds.
                = 5 cm × 12 = 60 cm
                
                ? Remaining string let out
                = (3 – 0.60) m
                = 2.4 m
            To find horizontal distance
            In the right ?PBC, let PB be the required horizonal distance of fly.
            Since, PB2 = PC2 – BC2
                PB2 = (2.4)2 – (1.8)2
                = 5.76 – 3.24 = 2.52
                
                Thus, the horizontal distance of the fly from Nazima after 12 seconds.
                = (1.59 + 1.2) m (approximately)
                = 2.76 m (approximately)
error: Content is protected !!