# Mathematics Questions and Answers – Criteria for Similarity of Triangle

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This set of Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Criteria for Similarity of Triangle”.

1. In the given figure DE∥BC, if AD = 5.9cm, DB = 4cm and AE = 7cm then, what will be the value of AC? a) 2.3 cm
b) 5.1 cm
c) 11.74 cm
d) 10.9 cm

Explanation: In ∆ABC DE∥BC, by Thales Theorem,
∴ $$\frac {AD}{DB}=\frac {AE}{EC}$$
$$\frac {5.9}{4}=\frac {7}{EC}$$
EC = $$\frac {7\times4}{5.9}$$ = 4.74 cm
AC = EC + AE = 7 + 4.74 = 11.74 cm

2. What is the value of x? a) 0
b) 1
c) 2
d) 3

Explanation: In ∆ABC DE∥BC, by Thales Theorem,
∴ $$\frac {AD}{DB}=\frac {AE}{EC}$$
$$\frac {x + 1}{3x – 1}=\frac {x}{4x – 3}$$
(x + 1)(4x – 3) = x(3x – 1)
4x2 – 3x + 4x – 3 = 3x2 – x
x2 + 2x – 3 = 0
x2 + 3x – x – 3 = 0
x(x + 3) – 1(x + 3) = 0
(x + 3)(x – 1) = 0
x = -3, 1
Since lengths cannot be negative so x = 1

3. In ∆PQR, PS is the bisector of ∠P. If PQ = 6.4cm, PR = 14cm, QS = 6cm. What will be the length of PS and QR? a) 10.5 cm
b) 12.74 cm
c) 10.74 cm
d) 11.74 cm

Explanation: In ∆ABC DE∥BC, by Thales Theorem,
∴ $$\frac {AD}{DB}=\frac {AE}{EC}$$
$$\frac {5.9}{4}=\frac {7}{EC}$$
EC = $$\frac {7\times4}{5.9}$$ = 4.74cm
AC = EC + AE = 7 + 4.74 = 11.74 cm

4. The perimeters of two similar triangles ABC, PQR is 64 cm and 24 cm respectively. If PQ is 12 cm what will be the length of AB?
a) 30 cm
b) 32 cm
c) 12 cm
d) 16 cm

Explanation: We know that the ratio of the perimeters of similar triangles is the same as the ratio of their corresponding sides.
∴ $$\frac {Perimeter \, of \, \triangle ABC}{Perimeter \, of \, \triangle PQR} = \frac {AB}{PQ}$$
$$\frac {64}{24}=\frac {AB}{12}$$
AB = $$\frac {64\times 12}{24}$$ = 32 cm

5. In the given figure, LM∥PQ, what will be the relation between x, a, b and c? a) a = $$\frac {c}{b}$$
b) ab = cx
c) bx = ac
d) cb = ax

Explanation: LM∥PQ by Thales Theorem,
$$\frac {PN}{PM}=\frac {QN}{QL}$$
$$\frac {c}{x}=\frac {b}{a}$$
bx = ac

6. In two similar triangles ∆ABC and ∆DEF, AB = 15cm, DE = 5cm. If AL and DM are the altitudes of the triangles ABC, DEF respectively, then what will be the ratio of their altitudes? a) 3 : 1
b) 1 : 3
c) 1 : 2
d) 2 : 1

Explanation: We know that, if two triangles are equiangular, then the ratio of their corresponding sides is the same as the ratio of their corresponding altitudes.
Since, ∆ABC ∼ ∆DEF therefore, the corresponding angles will be equal.
∴ $$\frac {AB}{DE}=\frac {AL}{DM}$$
$$\frac {15}{5}=\frac {AL}{DM}$$
$$\frac {3}{1}=\frac {AL}{DM}$$

7. If DE∥BC, AD = 4cm, BD = 7.5cm, AE = 6.4 cm & DE = 5cm then what will be the lengths of AC and BC? a) 11.23 cm
b) 15.24 cm
c) 14.37 cm
d) 14.25 cm

Explanation: In ∆ABC DE∥BC, by Thales Theorem,
∴ $$\frac {AD}{DB}=\frac {AE}{EC}$$
$$\frac {4}{7}=\frac {6.4}{EC}$$
EC = $$\frac {7\times 6.4}{4}$$ = 11.2 cm
AC = AE + EC = 6.4 + 11.2 = 17.6cm
∠A = ∠D     (Common angle)
∠AEB = ∠ACB     (Corresponding angles)
∴ ∆ABC ∼ ∆ADE     (AA similarity)
∴ $$\frac {AD}{AB}=\frac {DE}{BC}$$
$$\frac {4}{7.5 + 4} = \frac {5}{BC}$$
BC = $$\frac {11.5\times 5}{4}$$ = 14.37 cm

8. If ∠A = ∠P, ∠B = ∠Q, ∠C = ∠R then, ∆ABC & ∆PQR are similar according to which test?
a) AAA test
b) AA test
c) SAS test
d) SSS test

Explanation: Since, the angles between the two triangles are equal; therefore the two triangles are similar according to the Angle – Angle – Angle test of similarity. 9. If ∠D = ∠L, ∠E = ∠M then, ∆DEF & ∆LMN are similar according to which test?
a) AAA test
b) AA test
c) SAS test
d) SSS test

Explanation: Since, the two angles of the triangles are equal; therefore the triangles are similar according to the Angle – Angle test of similarity. 10. If $$\frac {AB}{XY} = \frac {BC}{YZ} = \frac {AC}{XZ}$$ then, ∆ABC & ∆XYZ are similar according to which test?
a) AAA test
b) AA test
c) SAS test
d) SSS test

Explanation: Since, the sides of the triangle are proportional to each other; therefore the triangles are similar according to the Side – Side – Side test of similarity. Here, the sides of the triangle are in proportional to each other. Hence, they are similar according to the SSS test.

Sanfoundry Global Education & Learning Series – Mathematics – Class 10.

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