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Area Theorems [Proof and Use] MATHEMATICS - Area Theorems [Proof and Use]

ICSE - STD 9 - MATHEMATICS (ICSE - English Medium). 56 questions in this chapter.

Question types

Area Theorems [Proof and Use] question types

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

56
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4
Question groups
5
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Sample Questions

Area Theorems [Proof and Use] questions

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

Q 1[2 Mark Question Answer]2 Marks
In the following figure$, \text{CE}$ is drawn parallel to diagonals $\text{DB}$ of the quadrilateral $\text{ABCD}$ which meets $AB$ produced at point $E.$Prove that $\triangle ADE$ and quadrilateral $\text{ABCD}$ are equal in area.
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Q 2[2 Mark Question Answer]2 Marks
In the given figure $, D$ is mid $-$ point of side $\text{AB}$ of $\triangle ABC$ and $\text{BDEC}$ is a parallelogram.

Prove that$:$ Area of $ABC =$ Area of $\| gm \ \text{BDEC}.$
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Q 3[3 marks sum]3 Marks
In the following figure, $OAB$ is a triangle and $AB \| DC.$

If the area of $\triangle CAD=140 \ cm^2$ and the area of $\triangle ODC=172\ \ cm^2$, find:$(i)$ the area of $\triangle DBC,(ii)$ the area of $\triangle OAC,(iii)$ the area of $\triangle ODB$.
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Q 4[3 marks sum]3 Marks
In the following figure, $BD$ is parallel to $CA, E$ is mid$-$point of $CA$ and $BD = `1/2`CA$.Prove that: $ar. ( \triangle ABC ) = 2 \times ar.( \triangle DBC )$
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Q 5[3 marks sum]3 Marks
The perimeter of a $\triangle ABC$ is $37 \ cm$ and the ratio between the lengths of its altitudes be $6: 5: 4.$ Find the lengths of its sides. Let the sides be $x \ cm, y \ cm$ and $(37 - x - y) \ cm.$ Also, let the lengths of altitudes be $6a \ cm, 5a \ cm$ and $4a \ cm.$
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Q 6[3 marks sum]3 Marks
The base $BC$ of $\triangle ABC$ is divided at $D$ .so that $BD=\frac{1}{2} DC$.Prove that area of $ΔABD=\frac{1}{3}$ of the area of $ΔABC.$
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Q 7[3 marks sum]3 Marks
In the figure of question $2,$ if $E$ is the mid$-$ point of median $AD,$ then prove that:Area $( ΔABE ) =\frac{1}{4}$ Area $( ΔABC ).$
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Q 8[4 marks sum]4 Marks
$\text{ABCD}$ is a trapezium with $AB$ parallel to $DC. A$ line parallel to $AC$ intersects $AB$ at $X$ and $BC$ at $Y.$
Prove that the area of $\triangle ADX =$ area of $\triangle ACY.$
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Q 10[4 marks sum]4 Marks
In the given figure, $AP$ is parallel to $BC, BP$ is parallel to $CQ$.Prove that the area of triangles $\text{ABC}$ and $\text{BQP}$ are equal.
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Q 11[4 marks sum]4 Marks
The given figure shows a pentagon $\text{ABCDE}. EG$ drawn parallel to $DA$ meets $BA$ produced at $G$ and $CF$ draw parallel to$ DB$ meets $AB$ produced at $F.$
Prove that the area of pentagon $\text{ABCDE}$ is equal to the area of $\triangle GDF.$
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Q 12[4 marks sum]4 Marks
$\text{ABCD}$ is a parallelogram a line through $A$ cuts $DC$ at point $P$ and $BC$ produced at $Q.$ Prove that $\triangle BCP$ is equal in area to $\triangle DPQ.$
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Q 13[5 marks sum]5 Marks
In parallelogram $\text{ABCD, E}$ is a point in $AB$ and $DE$ meets diagonal $AC$ at point $F$. If $DF: FE = 5:3$ and area of $\triangle ADF$ is $60 \ cm^2$; find,$(i)$ area of $\triangle ADE.(ii)$ if $AE: EB = 4:5$, find the area of $\triangle ADB.(iii)$ also, find the area of parallelogram $\text{ABCD}.$
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Q 14[5 marks sum]5 Marks
The medians of a $\triangle ABC$ intersect each other at point $G$. If one of its medians is $AD$,prove that:$(i)$ Area $( \triangle ABD ) = 3 \times$ Area $( \triangle BGD );(ii)$ Area $( \triangle ACD ) = 3 \times$ Area $( \triangle CGD )(iii)$ Area $( \triangle BGC ) = \frac{1}{3} \times$ Area $( \triangle ABC ).$
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Q 15[5 marks sum]5 Marks
In parallelogram $\text{ABCD, P}$ is the mid$-$point of $AB. CP$ and $BD$ intersect each other at point $O$. If the area of $\triangle POB = 40 \ cm^2$, and $OP: OC = 1:2,$ find:$(i)$ Areas of $\triangle BOC$ and $\triangle PBC;(ii)$ Areas of $\triangle ABC$ and parallelogram $\text{ABCD}.$
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Q 16[5 marks sum]5 Marks
In $ΔABC, E$ and $F$ are mid$-$points of sides $AB$ and $AC$ respectively. If $BF$ and $CE$ intersect each other at point $O$,prove that the $ΔOBC$ and quadrilateral $\text{AEOF}$ are equal in area.
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Q 17[5 marks sum]5 Marks
The given figure shows a parallelogram $\text{ABCD}$ with area $324 sq. \ cm. P$ is a point in $AB$ such that $AP: PB = 1:2$.Find The area of $\triangle APD.$
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