### MCQ on Areas of parallelograms and Triangles

**Ques.** A parallelogram *ABCD* and a rectangle *ABPQ* are on the same base *AB* and between the same parallels *AB* and *CQ*. If *AB* = 8 cm and *AQ = *6 cm, find the area of ||gm *ABCD*.

(a) 52 cm^{2}

(b) 46 cm^{2}

(c) 48 cm^{2}

(d) 58 cm^{2}

Ans. (c)

**Ques. **In a parallelogram ABCD ∠D = 60^{o} then the measurement of ∠A

(a) 120^{o}

(b) 65^{o}

(c) 90^{o}

(d) 75^{o}

Ans. (a)

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**Ques. **In **△**ABC, AD is a median and P is a point is AD such that AP : PD = 1 : 2 then the area of **△**ABP =

(a) 1/3 x Area of **△**ABC

(b) 2/3 x Area of **△**ABC

(c) 1/3 x Area of **△**ABC

(d) 1/6 x Area of **△**ABC

Ans. (d)

**Ques. **Find the area a rhombus, the lengths of whose diagonals are 16 cm and 24 cm respectively.

(a) 92 cm2

(b) 192 cm2

(c) 108 cm2

(d) 168 cm2

Ans. (b)

**Ques. **The sum of the interior angles of polygon is three times the sum of its exterior angles. Then numbers of sides in polygon is

(a) 6

(b) 7

(c) 8

(d) 9

Ans. (c)

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**Ques. ***ABCD* is parallelogram. If *AB* = 3.6 cm and altitude corresponding to sides *AB* and *AD* are respectively 5 cm and 4 cm, then *AD* will be

(a) 5.5 cm

(b) 3.5 cm

(c) 2.5 cm

(d) 4.5 cm

Ans. (d)

**Ques. **In **△**ABC if D is a point in BC and divides it the ratio 3 : 5 i.e., if BD : DC = 3 : 5 then, ar (**△**ADC) : ar(**△**ABC) = ?

(a) 3 : 5

(b) 3 : 8

(c) 5 : 8

(d) 8 : 3

Ans. (c)

**Ques. **Find the area of trapezium whose parallel sides are 8 cm and 6 cm respectively and the distance between these sides is 8 cm.

(a) 56 cm^{2}

(b) 52 cm^{2}

(c) 58 cm^{2}

(d) 54 cm^{2}

Ans. (a)

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**Ques.** The area of trapezium whose parallel sides are 9cm and 6cm respectively and distance between these sides is 8 cm will be

(a) 50 cm^{2}

(b) 60 cm^{2}

(c) 70 cm^{2}

(d) 80 cm^{2}

Ans. (b)

**Ques. **Area of rhombus whose diagonals are 16cm and 24cm will be

(a) 182 cm^{2}

(b) 202 cm^{2}

(c) 92 cm^{2}

(d) 192 cm^{2}

Ans. (d)

**Ques.** In a parallelogram ABCD, AB = 8 cm. The altitudes corresponding to sides AB and AD are respectively 4 m and 5 cm. Find AD.

(a) 6.4 cm

(b) 6.6 cm

(c) 6.5 cm

(d) 6.2 cm

Ans. (a)

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**Subjective Questions**

- Prove: A diagonal of parallelogram divides it into two triangles of equal area.
*E, F, G, H*are respectively, the mid-points of the sides*AB, BC, CD*and*DA*of parallelogram*ABCD*. Show that the quadrilateral*EFGH*is a parallelogram and its area is half the area of the parallelogram*ABCD*.- Show that the diagonals of a parallelogram divide it into four triangles of equal area.
- In figure,
*PQRS*and*ABRS*are parallelograms and*X*is any point on the side*BR*. Show that

(i) ar. (*PQRS*) = ar. (*ABRS*)

(ii) ar. (AXS) = ½ ar. (PQRS)

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- Prove that parallelogram on the same base and between same parallel are equal in area.
- ABC is a triangle in which D is the mid-point of BC and E is the mid-point of AD. Prove that the area of
**△**BED=1/4 area of**△**ABC. - Prove: The area of a trapezium is half the product of its height and the sum of the parallel sides.
- In a triangle
*ABC*,*E*is any point on its median*AD*, show that ar. (*ABE*) = ar. (*ACE*). - Prove that parallelogram and a rectangle on the same base and between the same parallels are equal in area.
- Triangles ABC and DBC are on the same base BC; with A, D on opposite sides of the line BC, such that ar(
**△**ABC) = ar(**△**DBC). Show that BC bisects AD.

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- Parallelogram
*ABCD*and rectangle*ABEF*have the same base*AB*and also have equal areas. Show that the perimeter of the parallelogram is greater than that of the rectangle. - The diagonals of a parallelogram ABCD intersect in O. A line through O meets AB is X and the opposite side CD in Y. Show that ar (quadrilateral AXYD) = ½ far(parallelogram ABCD).
- Prove: Triangles having equal areas and having one side of the triangle equal to corresponding side of the other, have their corresponding altitudes equal.
- A farmer was having a field in the form of a ||gm
*PQRS*. He took any point*A*on*RS*and joined it to points*P*and*Q*. In how many parts the field is divided? What are the shapes of these parts? The farmer wants to sow wheat and pulses in equal portions of the field separately. How should he do it? - If a triangle and a parallelogram are on the same base and between the same parallels, then prove that the area of the triangle is equal to half the area of the parallelogram.

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- A
**△**ABC in which D is the mid-point of BC and E is the mid-point of AD. *P*and*Q*are any points lying on the sides*DC*and*AD*respectively of a parallelogram*ABCD*. Show that ar. (**△***APB*) = ar. (**△***BQC*).- Diagonals AC and BD of a quadrilateral ABCD intersect each other at P. Show that : ar(APB) + ar(CPD) = ar(APD) x ar(BPC)
- Prove: Two triangles on the same base (or equal bases) and between the same parallels are equal in area.
- Prove: The area of parallelogram is the product of its base and the corresponding altitude.
*ABCD*is a parallelogram.*X*and*Y*are the mid-points of*BC*and*CD*respectively. Prove that ar. (**△***AXY*) = 3/8 ar. (||gm*ABCD*).

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- ABCD is a quadrilateral and BD is one of its diagonal as shown in the figure. Show that the quadrilateral ABCD is a parallelogram and find its area.
- The diagonals of a parallelogram
*ABCD*intersect at a point*O*. Through*O*, a line is drawn to intersect*AD*at*P*and*BC*at*Q*. Show that*PQ*divides the parallelogram into two parts of equal area. - If each diagonal of a quadrilateral separates into two triangles of equal area, then show that the quadrilateral is a parallelogram.
- P and Q are any two points lying on the sides DC and AD respectively of parallelogram ABCD. Prove that : ar (
**△**APB) = ar(**△**BQC).

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