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06
NCERT Solutions · Maths Ch 6
Triangles
44 questions · 3 exercises
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Ex 6.1
Ex 6.2
Ex 6.3
NCERT answers with full exam-style working, step by step. Tap a question to open its solution.
1
7 QUESTIONS
Exercise 6.1
Q1(i)
Fill in the blank using the correct word given in brackets: All circles are ______. (congruent, similar)
(1 mark)
Q1(ii)
Fill in the blank using the correct word given in brackets: All squares are ______. (similar, congruent)
(1 mark)
Q1(iii)
Fill in the blank using the correct word given in brackets: All ______ triangles are similar. (isosceles, equilateral)
(1 mark)
Q1(iv)
Fill in the blanks using the correct word given in brackets: Two polygons of the same number of sides are similar, if (a) their corresponding angles are ______ and (b) their corresponding sides are ______. (equal, proportional)
(1 mark)
Q2(i)
Give two different examples of a pair of similar figures.
(1 mark)
Q2(ii)
Give two different examples of a pair of non-similar figures.
(1 mark)
Q3
State whether the following quadrilaterals are similar or not.
Figure (Fig. 6.8): PQRS is a rhombus with every side 1.5 cm and its angles NOT right angles (it is slanted). ABCD is a square with every side 3 cm and all angles 90°.
(1 mark)
2
13 QUESTIONS
Exercise 6.2
Q1(i)
In Fig. 6.17 (i), DE ∥ BC. Find EC.
Figure: △ABC with D on AB and E on AC, DE ∥ BC. AD = 1.5 cm, DB = 3 cm, AE = 1 cm.
(2 marks)
Q1(ii)
In Fig. 6.17 (ii), DE ∥ BC. Find AD.
Figure: △ABC with D on AB and E on AC, DE ∥ BC. DB = 7.2 cm, AE = 1.8 cm, EC = 5.4 cm.
(2 marks)
Q2(i)
E and F are points on the sides PQ and PR respectively of a △PQR. State whether EF ∥ QR: PE = 3.9 cm, EQ = 3 cm, PF = 3.6 cm and FR = 2.4 cm
(2 marks)
Q2(ii)
E and F are points on the sides PQ and PR respectively of a △PQR. State whether EF ∥ QR: PE = 4 cm, QE = 4.5 cm, PF = 8 cm and RF = 9 cm
(2 marks)
Q2(iii)
E and F are points on the sides PQ and PR respectively of a △PQR. State whether EF ∥ QR: PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and PF = 0.36 cm
(2 marks)
Q3
In Fig. 6.18, if LM ∥ CB and LN ∥ CD, prove that AM/AB = AN/AD.
Figure: quadrilateral ABCD with diagonal AC. L is a point on AC. M is on AB with LM ∥ CB, and N is on AD with LN ∥ CD.
(3 marks)
Q4
In Fig. 6.19, DE ∥ AC and DF ∥ AE. Prove that BF/FE = BE/EC.
Figure: △ABC with D on AB, and E and F on BC (F between B and E). DE ∥ AC and DF ∥ AE.
(3 marks)
Q5
In Fig. 6.20, DE ∥ OQ and DF ∥ OR. Show that EF ∥ QR.
Figure: △PQR with a point O inside it, joined to P, Q and R. D is on PO, E is on PQ and F is on PR.
(3 marks)
Q6
In Fig. 6.21, A, B and C are points on OP, OQ and OR respectively such that AB ∥ PQ and AC ∥ PR. Show that BC ∥ QR.
Figure: △PQR with a point O inside it, joined to P, Q and R. A is on OP, B on OQ and C on OR.
(3 marks)
Q7
Using 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.
(3 marks)
Q8
Using Theorem 6.2, prove that the line joining the mid-points of any two sides of a triangle is parallel to the third side.
(3 marks)
Q9
ABCD is a trapezium in which AB ∥ DC and its diagonals intersect each other at the point O. Show that AO/BO = CO/DO.
(3 marks)
Q10
The diagonals of a quadrilateral ABCD intersect each other at the point O such that AO/BO = CO/DO. Show that ABCD is a trapezium.
(3 marks)
3
24 QUESTIONS
Exercise 6.3
Q1(i)
State whether the pair of triangles in Fig. 6.34 is similar. Write the similarity criterion used and write the pair of similar triangles in symbolic form.
Figure: △ABC with ∠A = 60°, ∠B = 80°, ∠C = 40°; △PQR with ∠P = 60°, ∠Q = 80°, ∠R = 40°.
(1 mark)
Q1(ii)
State whether the pair of triangles in Fig. 6.34 is similar. Write the similarity criterion used and write the pair of similar triangles in symbolic form.
Figure: △ABC with AB = 2, BC = 2.5, CA = 3; △PQR with PQ = 6, QR = 4, PR = 5.
(1 mark)
Q1(iii)
State whether the pair of triangles in Fig. 6.34 is similar. Write the similarity criterion used and write the pair of similar triangles in symbolic form.
Figure: △LMP with LM = 2.7, LP = 3, MP = 2; △DEF with DE = 4, DF = 6, EF = 5.
(1 mark)
Q1(iv)
State whether the pair of triangles in Fig. 6.34 is similar. Write the similarity criterion used and write the pair of similar triangles in symbolic form.
Figure: △MNL with MN = 2.5, ML = 5, ∠M = 70°; △PQR with QP = 5, QR = 10, ∠Q = 70°.
(1 mark)
Q1(v)
State whether the pair of triangles in Fig. 6.34 is similar. Write the similarity criterion used and write the pair of similar triangles in symbolic form.
Figure: △ABC with AB = 2.5, BC = 3, ∠A = 80°; △DEF with DF = 5, EF = 6, ∠F = 80°.
(1 mark)
Q1(vi)
State whether the pair of triangles in Fig. 6.34 is similar. Write the similarity criterion used and write the pair of similar triangles in symbolic form.
Figure: △DEF with ∠D = 70°, ∠E = 80°; △PQR with ∠Q = 80°, ∠R = 30°.
(1 mark)
Q2
In Fig. 6.35, △ODC ~ △OBA, ∠BOC = 125° and ∠CDO = 70°. Find ∠DOC, ∠DCO and ∠OAB.
Figure: two parallel lines, with D and C on the top line and A and B on the bottom line. AC and BD cross at O. ∠CDO = 70° and ∠BOC = 125°.
(3 marks)
Q3
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 OA/OC = OB/OD.
(3 marks)
Q4
In Fig. 6.36, QR/QS = QT/PR and ∠1 = ∠2. Show that △PQS ~ △TQR.
Figure: △PQR with S on QR. T is a point such that Q, P, T are in a line (T beyond P) and T is joined to R. ∠1 is ∠PQR at Q, and ∠2 is ∠PRQ at R.
(3 marks)
Q5
S and T are points on sides PR and QR of △PQR such that ∠P = ∠RTS. Show that △RPQ ~ △RTS.
(3 marks)
Q6
In Fig. 6.37, if △ABE ≅ △ACD, show that △ADE ~ △ABC.
Figure: △ABC with D on AB and E on AC. BE and CD are joined, and so is DE.
(3 marks)
Q7(i)
In Fig. 6.38, altitudes AD and CE of △ABC intersect each other at the point P. Show that: △AEP ~ △CDP
Figure: △ABC with D on BC (AD ⊥ BC) and E on AB (CE ⊥ AB). AD and CE meet at P.
(2 marks)
Q7(ii)
In Fig. 6.38, altitudes AD and CE of △ABC intersect each other at the point P. Show that: △ABD ~ △CBE
Figure: △ABC with D on BC (AD ⊥ BC) and E on AB (CE ⊥ AB). AD and CE meet at P.
(2 marks)
Q7(iii)
In Fig. 6.38, altitudes AD and CE of △ABC intersect each other at the point P. Show that: △AEP ~ △ADB
Figure: △ABC with D on BC (AD ⊥ BC) and E on AB (CE ⊥ AB). AD and CE meet at P.
(2 marks)
Q7(iv)
In Fig. 6.38, altitudes AD and CE of △ABC intersect each other at the point P. Show that: △PDC ~ △BEC
Figure: △ABC with D on BC (AD ⊥ BC) and E on AB (CE ⊥ AB). AD and CE meet at P.
(2 marks)
Q8
E is a point on the side AD produced of a parallelogram ABCD and BE intersects CD at F. Show that △ABE ~ △CFB.
(3 marks)
Q9
In Fig. 6.39, ABC and AMP are two right triangles, right angled at B and M respectively. Prove that: (i) △ABC ~ △AMP (ii) CA/PA = BC/MP
Figure: A, B, P lie on a horizontal line, with B between A and P. C is above B (CB ⊥ AP). M is on AC with PM ⊥ AC.
(3 marks)
Q10
CD and GH are respectively the bisectors of ∠ACB and ∠EGF such that D and H lie on sides AB and FE of △ABC and △EFG respectively. If △ABC ~ △FEG, show that: (i) CD/GH = AC/FG (ii) △DCB ~ △HGE (iii) △DCA ~ △HGF
(5 marks)
Q11
In Fig. 6.40, E is a point on side CB produced of an isosceles triangle ABC with AB = AC. If AD ⊥ BC and EF ⊥ AC, prove that △ABD ~ △ECF.
Figure: isosceles △ABC with AB = AC. BC is extended beyond B to E. D is on BC with AD ⊥ BC. F is on AC with EF ⊥ AC.
(3 marks)
Q12
Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of △PQR (see Fig. 6.41). Show that △ABC ~ △PQR.
Figure: △ABC with median AD (D is the mid-point of BC) and △PQR with median PM (M is the mid-point of QR).
(5 marks)
Q13
D is a point on the side BC of a triangle ABC such that ∠ADC = ∠BAC. Show that CA² = CB.CD.
(3 marks)
Q14
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.
(5 marks)
Q15
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.
(3 marks)
Q16
If AD and PM are medians of triangles ABC and PQR, respectively where △ABC ~ △PQR, prove that AB/PQ = AD/PM.
(3 marks)