class 10 maths Triangles Hots Questions

Q 1 :
If $∆$ ABC ~ $∆$ EDF and $∆$ ABC is not similar to $∆$ DEF, then which of the following is not true ?

BC. EF = AC. FD
AB. EF = AC. DE
BC. DE = AB. EF
BC. DE = AB. FD

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(3)

Since, $∆$ ABC ~ $∆$ EDF

Therefore, the ratio of their corresponding sides is proportional.

$\Rightarrow \frac{B C}{E F} = \frac{A B}{D E} \Rightarrow B C . D E \neq A B . E F$

Q 2 :
In the $∆$ ABC, DE $| |$ BC and AD = 3x − 2, AE = 5x − 4, BD = 7x − 5, CE = 5x − 3, then find the value of x

1
7/10
both (a) & (b)
none of these

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2

3

4

(3)

Given that, AD = 3x − 2, AE = 5x − 4, BD = 7x − 5, CE = 5x − 3

By Basic Proportionality theorem, we have $\frac{A D}{B D} = \frac{A E}{E C}$

$\Rightarrow \frac{3 x - 2}{7 x - 5} = \frac{5 x - 4}{5 x - 3} \Rightarrow (3 x - 2) (5 x - 3) = (5 x - 4) (7 x - 5)$

$\Rightarrow 15 x^{2} - 19 x + 6 = 35 x^{2} - 53 x + 20$

$\Rightarrow 20 x^{2} - 34 x + 14 = 0 \Rightarrow 10 x^{2} - 17 x + 7 = 0$

$\Rightarrow (x - 1) (10 x - 7) = 0 \Rightarrow x = 1, x = \frac{7}{10}$

Q 3 :
In the given figure, DE $∥$ BC, AE = a units, EC = b units, DE = x units and BC = y units. Which of the following is true?

[IMAGE]

$x = \frac{a + b}{a y}$
$y = \frac{a x}{a + b}$
$x = \frac{a y}{a + b}$
$\frac{x}{y} = \frac{a}{b}$

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(3)

[IMAGE]

As $D E ∥ B C$

$∴$ $\frac{A E}{A C} = \frac{D E}{B C}$ $[From BPT]$

$\Rightarrow \frac{a}{a + b} = \frac{x}{y}$

$\Rightarrow x (a + b) = a y \Rightarrow x = \frac{a y}{a + b}$

Q 4 :
ABCD is a trapezium with AD $∥$ BC and AD = 4cm. If the diagonals AC and BD intersect each other at O such that AO/OC = DO/OB =1/2, then BC =

6 cm
7 cm
8 cm
9 cm

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(3)

[IMAGE]

$A D ∥ B C$

and $\frac{A O}{O C} = \frac{D O}{O B} = \frac{1}{2}$

$∴$ $\frac{A D}{B C} = \frac{A O}{O C} \Rightarrow \frac{4}{B C} = \frac{1}{2} \Rightarrow B C = 8 cm$

Q 5 :
$ΔABC ~ ΔPQR$ . If AM and PN are altitudes of $ΔABC$ and $ΔPQR$ respectively and $A B^{2} : P Q^{2}$ = 4 : 9, then AM : PN =

3 : 2
16 : 81
4 : 9
2 : 3

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(4)

[Image]

We have, $∆ A B C ~ ∆ P Q R$

$\Rightarrow \frac{A B}{P Q} = \frac{B C}{Q R} = \frac{C A}{R P} = \frac{A M}{P N} (corresponding sides of similar triangles)$

But $\frac{A B^{2}}{P Q^{2}} = \frac{4}{9} \Rightarrow \frac{A B}{P Q} = \frac{2}{3}$

i.e., $\frac{A B}{P Q} = \frac{A M}{P N} = \frac{2}{3}$

Hence, $A M : P N = 2 : 3$

Q 6 :
The perimeters of two similar triangles ABC and PQR are 56 cm and 48 cm respectively. PQ/AB is equal to

7/8
6/7
7/6
8/7

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4

(2)

The ratio of the corresponding sides of similar triangles is same as the ratio of their perimeter

$∴$ $∆ A B C ~ ∆ P Q R or ∆ P Q R ~ ∆ A B C$

$\Rightarrow \frac{P Q}{A B} = \frac{Q R}{B C} = \frac{P R}{A C} = \frac{48}{56} \Rightarrow \frac{P Q}{A B} = \frac{6}{7}$

Q 7 :
In the given figure, if M and N are points on the sides OP and OS respectively of $∆ O P S$ , such that MN $∥$ PS, then the length of OP is :

[IMAGE]

6.8 cm
17 cm
15.3 cm
9.6 cm

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(3) 15.3 cm

Q 8 :
In the given figure $∆ A B C$ is shown. DE is parallel to BC. If AD = 5 cm, DB = 2.5 cm and BC = 12 cm, then DE is equal to

[IMAGE]

10 cm
6 cm
8 cm
7.5 cm

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(3)

$A D = 5 cm$

$D B = 2.5 cm$

$B C = 12 cm$

$D E ∥ B C$

$∆ A B C ~ ∆ A D E$

$\Rightarrow \frac{A D}{A B} = \frac{D E}{B C} \Rightarrow \frac{5}{7.5} = \frac{D E}{12} \Rightarrow D E = \frac{60}{7.5} = 8 cm$

Topic Question Set

Q 1 :
If $∆$ ABC ~ $∆$ EDF and $∆$ ABC is not similar to $∆$ DEF, then which of the following is not true ?

Q 2 :
In the $∆$ ABC, DE $| |$ BC and AD = 3x − 2, AE = 5x − 4, BD = 7x − 5, CE = 5x − 3, then find the value of x

Q 3 :
In the given figure, DE $∥$ BC, AE = a units, EC = b units, DE = x units and BC = y units. Which of the following is true?

[IMAGE]

Q 4 :
ABCD is a trapezium with AD $∥$ BC and AD = 4cm. If the diagonals AC and BD intersect each other at O such that AO/OC = DO/OB =1/2, then BC =

Q 5 :
$ΔABC ~ ΔPQR$ . If AM and PN are altitudes of $ΔABC$ and $ΔPQR$ respectively and $A B^{2} : P Q^{2}$ = 4 : 9, then AM : PN =

Q 6 :
The perimeters of two similar triangles ABC and PQR are 56 cm and 48 cm respectively. PQ/AB is equal to

Q 7 :
In the given figure, if M and N are points on the sides OP and OS respectively of $∆ O P S$ , such that MN $∥$ PS, then the length of OP is :

[IMAGE]

Q 8 :
In the given figure $∆ A B C$ is shown. DE is parallel to BC. If AD = 5 cm, DB = 2.5 cm and BC = 12 cm, then DE is equal to

[IMAGE]

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Topic Question Set

Q 1 : If ∆ABC ~ ∆EDF and ∆ABC is not similar to ∆DEF, then which of the following is not true ?

Q 2 : In the ∆ABC, DE || BC and AD = 3x − 2, AE = 5x − 4, BD = 7x − 5, CE = 5x − 3, then find the value of x

Q 3 : In the given figure, DE ∥ BC, AE = a units, EC = b units, DE = x units and BC = y units. Which of the following is true? [IMAGE]

Q 4 : ABCD is a trapezium with AD∥BC and AD = 4cm. If the diagonals AC and BD intersect each other at O such that AO/OC = DO/OB =1/2, then BC =

Q 5 : ΔABC~ΔPQR. If AM and PN are altitudes of ΔABC and ΔPQR respectively and AB2:PQ2 = 4 : 9, then AM : PN =

Q 6 : The perimeters of two similar triangles ABC and PQR are 56 cm and 48 cm respectively. PQ/AB is equal to

Q 7 : In the given figure, if M and N are points on the sides OP and OS respectively of ∆OPS, such that MN∥PS, then the length of OP is : [IMAGE]

Q 8 : In the given figure ∆ABC is shown. DE is parallel to BC. If AD = 5 cm, DB = 2.5 cm and BC = 12 cm, then DE is equal to [IMAGE]

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Q 1 :
If $∆$ ABC ~ $∆$ EDF and $∆$ ABC is not similar to $∆$ DEF, then which of the following is not true ?

Q 2 :
In the $∆$ ABC, DE $| |$ BC and AD = 3x − 2, AE = 5x − 4, BD = 7x − 5, CE = 5x − 3, then find the value of x

Q 3 :
In the given figure, DE $∥$ BC, AE = a units, EC = b units, DE = x units and BC = y units. Which of the following is true?

[IMAGE]

Q 4 :
ABCD is a trapezium with AD $∥$ BC and AD = 4cm. If the diagonals AC and BD intersect each other at O such that AO/OC = DO/OB =1/2, then BC =

Q 5 :
$ΔABC ~ ΔPQR$ . If AM and PN are altitudes of $ΔABC$ and $ΔPQR$ respectively and $A B^{2} : P Q^{2}$ = 4 : 9, then AM : PN =

Q 6 :
The perimeters of two similar triangles ABC and PQR are 56 cm and 48 cm respectively. PQ/AB is equal to

Q 7 :
In the given figure, if M and N are points on the sides OP and OS respectively of $∆ O P S$ , such that MN $∥$ PS, then the length of OP is :

[IMAGE]

Q 8 :
In the given figure $∆ A B C$ is shown. DE is parallel to BC. If AD = 5 cm, DB = 2.5 cm and BC = 12 cm, then DE is equal to

[IMAGE]