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Similarity of Triangles Mathematics - Similarity of Triangles

ICSE - STD 10 - Mathematics (ICSE - English Medium). 38 questions in this chapter.

Question types

Similarity of Triangles question types

38 questions across 4 question groups — pick any mix to generate a Mathematics paper with step-by-step answer keys.

38
Questions
4
Question groups
5
Question types
Sample Questions

Similarity of Triangles questions

One sample from each question group in this chapter. Select any group above to see the full set with answer keys.

Q 1[3 marks sum]3 Marks
ABC is a right angled triangle with $\angle \mathrm{ABC}=90^{\circ} . \mathrm{D}$ is any point on AB and DE is perpendicular to AC , Prove that:
(a) $\triangle \mathrm{ADE} \sim \triangle \mathrm{ACB}$
(b) If $\mathrm{AC}=13 \mathrm{~cm}, \mathrm{BC}=5 \mathrm{~cm}$ and $\mathrm{AE}=4 \mathrm{~cm}$, find DE and AD .
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Q 2[3 marks sum]3 Marks
In $\triangle \mathrm{ABC}, \angle \mathrm{ABC}=\angle \mathrm{DAC}, \mathrm{AB}=8 \mathrm{~cm}$. $\mathrm{AC}=4 \mathrm{~cm}$ and $\mathrm{AD}=5 \mathrm{~cm}$
(a) Prove that $\triangle \mathrm{ACD} \sim \triangle \mathrm{BCA}$
(b) Find BC and CD
(c) Find $\operatorname{ar}(\triangle \mathrm{ACD})$ : $\operatorname{ar}(\triangle \mathrm{ABC})$
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Q 3[3 marks sum]3 Marks
In the given figure $\triangle A B C$ and $\triangle A M P$ are right angled at $B$ and $M$ respectively.
Given $\mathrm{AC}=10 \mathrm{~cm}, \mathrm{AP}=15 \mathrm{~cm}$ and $\mathrm{PM}=12 \mathrm{~cm}$.
(a) Prove $\triangle \mathrm{ABC} \sim \triangle \mathrm{AMP}$
(b) Find AB and BC .
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Q 4[3 marks sum]3 Marks
In the adjoining figure, ABC is a right angled triangle with $\angle \mathrm{BAC}=90^{\circ}$ and $\mathrm{AD} \perp \mathrm{BC}$.
(a) Prove $\triangle \mathrm{ADB} \sim \triangle \mathrm{CDA}$.
(b) If $\mathrm{BD}=18 \mathrm{~cm}, \mathrm{CD}=8 \mathrm{~cm}$, find AD .
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Q 5[3 marks sum]3 Marks
In the given figure, PQRS is a cyclic quadrilateral, PQ and SR produced meet at T .
(a) Prove $\triangle \mathrm{TPS} \sim \Delta \mathrm{TRQ}$.
(b) Find SP if $\mathrm{TP}=18 \mathrm{~cm}, \mathrm{RQ}=4 \mathrm{~cm}$ and $\mathrm{TR}=6 \mathrm{~cm}$
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Q 6[4 marks sum]4 Marks
In a trapezium, $\mathrm{ABCD}, \mathrm{AB}=\frac{1}{2} \mathrm{CD}$. EF drawn parallel to AB cuts AD at F and BC at E , such that $\frac{\mathrm{BE}}{\mathrm{EC}}=\frac{3}{4}$. If diagonal BD intersects EF at G , find $\frac{\mathrm{EF}}{\mathrm{AB}}$
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Q 8[4 marks sum]4 Marks
In the figure, $\mathrm{DE} \| \mathrm{BC}$ and $\mathrm{AD}: \mathrm{DB}=4: 3$.
(a) Show that $\triangle \mathrm{ADE} \sim \triangle \mathrm{ABC}$
(b) Find $\frac{\mathrm{AD}}{\mathrm{AB}}$ and $\frac{\mathrm{DE}}{\mathrm{BC}}$
(c) Prove that $\operatorname{ar}(\triangle \mathrm{DEF}): \operatorname{ar}(\triangle \mathrm{DEC})=4: 11$
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Q 10[4 marks sum]4 Marks
In the figure, ABCD is a trapezium in which $\mathrm{AB} \| \mathrm{DC}$ and the diagonals AC and BD intersect at O . Prove that :
(a) $\triangle \mathrm{OCD} \sim \triangle \mathrm{OAB}$
(b) If $\mathrm{OA}=(2 x+1) \mathrm{cm}, \mathrm{OB}=(5 x-3) \mathrm{cm}, \mathrm{OC}=(6 x-5) \mathrm{cm}$ and $\mathrm{OD}=(3 x-1) \mathrm{cm}$, find the value of $x$.
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Q 11MCQ1 Mark
In the given figure, $\mathrm{DE} \| \mathrm{BC}$ and $\mathrm{AD}: \mathrm{BD}=5: 4$. $\operatorname{ar}(\triangle \mathrm{DEF}): \operatorname{ar}(\triangle \mathrm{CFB})$ is :
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  • A
    $25: 81$
  • B
    $16: 9$
  • C
    $9: 16$
  • D
    $64: 25$
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Q 12MCQ1 Mark
The perimeters of two similar triangles ABC and XYZ are 60 cm and 48 cm respectively. If $X Y=8 \mathrm{~cm}$, then the length of $A B$ is
  • A
    12 cm
  • B
    11 cm
  • C
    10 cm
  • D
    9 cm
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Q 14MCQ1 Mark
In the figure, if $\triangle \mathrm{ADE} \sim \triangle \mathrm{ABC}$, then the length of BC is :
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  • A
    3 cm
  • B
    3.2 cm
  • C
    3.8 cm
  • D
    4.5 cm
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Q 15MCQ1 Mark
M and N are respectively the points on the sides XY and XZ of a triangle XYZ such that $\mathrm{XM}=3 \mathrm{~cm}, \mathrm{MY}=5 \mathrm{~cm}$, and $\mathrm{YZ}=12 \mathrm{~cm}$. If $\mathrm{MN} \| \mathrm{YZ}$, then the length of MN is:
  • A
    6 cm
  • B
    5 cm
  • C
    4.5 cm
  • D
    4 cm
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Q 16Assertion (A) & Reason (B) MCQ1 Mark
Assertion (A) : In the figure, $\mathrm{DE} \| \mathrm{BC}, \mathrm{AD}=(4 x-3) \mathrm{cm}, \mathrm{AE}=(8 x-7) \mathrm{cm}$, $\mathrm{BD}=(3 x-1) \mathrm{cm}$ and $\mathrm{CE}=(5 x-3) \mathrm{cm}$. The value of $x$ is 2.
Reason (R) : In $\triangle A B C$, if $D E \| B C$, then $\frac{A D}{A B}=\frac{A E}{A C}$.
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  • A
    A is true, R is false
  • B
    A is false, R is true
  • ✓
    Both A and R are true, and R is the correct reason for A.
  • D
    Both A and R are true, and R is incorrect reason for A .

Answer: C.

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Q 17Assertion (A) & Reason (B) MCQ1 Mark
Assertion (A): Two congruent triangles are similar but two similar triangles need not be congruent.
Reason (R): The areas of two similar triangles are proportional to their corresponding sides.
  • A
    A is true, R is false
  • ✓
    A is false, R is true
  • C
    Both A and R are true, and R is the correct reason for A.
  • D
    Both A and R are true, and R is incorrect reason for A .

Answer: B.

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Q 18Assertion (A) & Reason (B) MCQ1 Mark
Assertion (A): The perimeters of two similar triangles are 25 cm and 15 cm respectively. If one side of first triangle is 9 cm, then the corresponding side of the second triangle is 5.4 cm.
Reason (R): The ratio of perimeters of two similar triangles is same as the ratio of their corresponding sides.
  • ✓
    A is true, R is false
  • B
    A is false, R is true
  • C
    Both A and R are true, and R is the correct reason for A.
  • D
    Both A and R are true, and R is incorrect reason for A .

Answer: A.

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