Similarity - ICSE STD 10 Mathematics Questions

Q 1[1 Mark Question Answer]1 Mark

On a map drawn to a scale of $1 : 2,50,000;$ a triangular plot of land has the following measurements $: AB = 3\ cm, BC = 4\ cm$ and angle $ABC = 90^\circ.$
Calculate : the area of the plot in sq. km.

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Q 2[1 Mark Question Answer]1 Mark

A model of an aeroplane is made to a scale of 1 : 400. Calculate :

the length, in m, of the aeroplane, if length of its model is 16 cm.

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Q 3[1 Mark Question Answer]1 Mark

A model of an aeroplane is made to a scale of 1 : 400. Calculate :
the length, in cm, of the model; if the length of the aeroplane is 40 m

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Q 4[1 Mark Question Answer]1 Mark

A triangle ABC has been enlarged by scale factor m = 2.5 to the triangle A' B' C' Calculate : the length of AB, if A' B' = 6 cm.

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Q 5[1 Mark Question Answer]1 Mark

Area of two similar triangles are 98 sq.cm and 128 sq.cm. Find the ratio between the lengths
of their corresponding sides.

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Q 6[2 Mark Question Answer]2 Marks

In the following figure, DE || Ac and DC || AP. Prove that:$\frac{B E}{E C}=\frac{B C}{C P}$

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Q 7[2 Mark Question Answer]2 Marks

In the given figure, lines l, m and n are such that ls ∥ m ∥ n. Prove that:
$\frac{A B}{B C}=\frac{P Q}{Q R}$

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Q 8[2 Mark Question Answer]2 Marks

In the following figure, ∠AXY = ∠AYX. If $\frac{B X}{A X}=\frac{C Y}{A Y}$ show that triangle ABC is isoscele

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Q 9[2 Mark Question Answer]2 Marks

Triangle ABC is similar to triangle PQR. If AD and PM are altitudes of two triangles, Prove that :
$\frac{A B}{P Q}=\frac{A D}{P M}$

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Q 10[2 Mark Question Answer]2 Marks

ABC is a right angled triangle with ∠ABC = 90°. D is any point on AB and DE is perpendicular to AC. Prove that

ΔADE ~ ΔACB.

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Q 11[3 marks sum]3 Marks

Triangle ABC is similar to triangle PQR. If bisector of angle BAC meets BC at point D and bisector of angle QPR meets QR at point M, prove that $\frac{A B}{P Q}=\frac{A D}{P M}$

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Q 12[3 marks sum]3 Marks

Triangle ABC is similar to triangle PQR. If AD and PM are corresponding medians of the two triangles, prove that:
$\frac{A B}{P Q}=\frac{A D}{P M}$

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Q 13[3 marks sum]3 Marks

Given $: AB || DE$ and $BC || EF.$ Prove that:
$\frac{A D}{D G}=\frac{C F}{F G}$
$\Delta DFG \sim \Delta ACG$

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Q 14[3 marks sum]3 Marks

$ABC$ is a right angled triangle with $\angle ABC = 90^\circ. D$ is any point on $AB$ and $DE$ is perpendicular to $AC.$ Prove that

Find, area of $\triangle ADE :$ area of quadrilateral $BCED.$

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Q 15[3 marks sum]3 Marks

$ABC$ is a right angled triangle with $\angle ABC = 90^\circ. D$ is any point on $AB$ and $DE$ is perpendicular to $AC.$ Prove that

If $AC = 13\ cm, BC = 5\ cm$ and $AE = 4\ cm.$ Find $DE$ and $AD.$

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Q 16[5 marks sum]5 Marks

In the following figure, ABCD to a trapezium with AB ‖ DC. If AB = 9 cm, DC = 18 cm, CF = 13.5,cm, AP = 6 cm and BE = 15 cm, Calculate: PE

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Q 17[5 marks sum]5 Marks

In Δ ABC, ∠ABC = ∠DAC. AB = 8 cm, AC = 4 cm, AD = 5 cm.
1) Prove that ΔACD is similar to ΔBCA.
2) Find BC and CD
3) Find area of ΔACD: area of ΔBCA

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Q 18[5 marks sum]5 Marks

In the figure, given below, $ABCD$ is a parallelogram. $P$ is a point on $BC$ such that $BP : PC = 1:$
2. $DP$ produced meets $AB$ produces at $Q.$ Given the area of triangle $CPQ = 20 cm^2$

Calculate:
(i) area of triangle $CDP,$
(ii) area of parallelogram $ABCD.$

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Q 19[5 marks sum]5 Marks

In the given triangle PQR, LM is parallel to QR and PM : MR = 3: 4

Calculate the value of ratio:
1) $\frac{P L}{P Q}$ and then $\frac{L M}{Q R}$
2) $\frac{\text { Area of } \Delta LMN }{\text { Area of } \Delta MNR }$
3) $\frac{\text { Area of } \Delta LQM }{\text { Area of } \Delta LQN }$

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Q 20[5 marks sum]5 Marks

The given figure shows a trapezium in which AB is parallel to DC and diagonals AC and BD
intersect at point P. If AP : CP = 3: 5,

Find:
(1) ∆APB : ∆CPB (2 ) ∆DPC : ∆APB
(3) ∆ADP : ∆APB (iv) ∆APB : ∆ADB

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Q 21[4 marks sum]4 Marks

In the following figure, $AB, CD$ and $EF$ are perpendicular to the straight line $BDF.$

If $AB = x; CD = z$ unit and $EF = y$ unit, prove that:
$\frac{1}{x}+\frac{1}{y}=\frac{1}{z}$

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Q 22[4 marks sum]4 Marks

PQR is a triangle. S is a point on the side QR of ΔPQR such that ∠PSR = ∠QPR. Given QP = 8 cm, PR = 6 cm and SR = 3 cm.1.
1. i. ProveΔPQR∼ Δ
2. Find the lengths of QR and PS.
3. $\frac{\text { Area of } \triangle P Q R}{\text { area of } \triangle S P R}$

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Q 23[4 marks sum]4 Marks

Two isosceles triangles have equal vertical angles. Show that the triangles are similar.If the ratio between the areas of these two triangles is $16 : 25,$ find the ratio between their corresponding altitudes.

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Q 24[4 marks sum]4 Marks

On a map, drawn to a scale of $1 : 20000,$ a rectangular plot of land $ABCD$ has $AB = 24\ cm$ and $BC = 32\ cm.$ Calculate:(i) the diagonal distance of the plot in kilometer
(ii) the area of the plot in sq.km

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Q 25[4 marks sum]4 Marks

The following figure shows a triangle PQR in which XY is parallel to QR. If $PX : XQ = 1 : 3$ and $QR = 9$ cm. find the length of XY.