### 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}

**Ques. **One side of a rectangular field is 15 m and one of its diagonals is 17 m. Find the area of the field.

(a) 120 m2

(b) 80 m2

(c) 90 m2

(d) 70 m2

**Ques.** The area of the parallelogram formed by the lines y = mx, y = mx + 1, y = nx and y = nx + 1 equals

(a) |m + n| / (m – n)^{2}

(b) 2 / | m + n |

(c) 1 / | m + n|

(d) 1 / |m – n|

**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}

**Ques. **Find the area of a square, one of whose diagonals is 3.8 m long.

(a) 8.25 m2

(b) 8.22 m2

(c) 7.25 m2

(d) 7.22 m2

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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

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

(a) 92 cm^{2}

(b) 192 cm^{2}

(c) 108 cm^{2}

(d) 168 cm^{2}

**Ques. **Find the cost of carpeting a room 13 m long and 9 m broad with a carpet 75 cm wide at the rate of $12.40 per square metre.

(a) 1944.40

(b) 1934.40

(c) 1930.40

(d) 1940.40

**Ques.** The resultant of three forces represented in magnitude and direction by the sides of a triangle *ABC* taken in order with *BC* = 5 *cm*, *CA* = 5 *cm*, and *AB* = 8 *cm*, is a couple of moment

(a) 12 unit

(b) 24 unit

(c) 36 unit

(d) 16 unit

**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

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**Ques. **A room is half as long again as it is broad. The cost of carpeting the at $5 per sq. m is $270 and the cost of papering the four walls at $10 per m2 is $1720. If a door and 2 windows occupy 8 sq. m, find the dimensions of the room.

(a) 19 m

(b) 13 m

(c) 7 m

(d) 6 m

**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

**Ques.** If the equations of opposite sides of a parallelogram are x^{2} – 7x + 6 = 0 and y^{2} – 14y + 40 = 0, then the equation of its one diagonal is

(a) 6x + 5y + 14 = 0

(b) 6x – 5y + 14 = 0

(c) 5x + 6y + 14 = 0

(d) 5x – 6y + 14 = 0

**Ques.** The sides of a triangle are in the ratio 3 : 5 : 7 and its perimeter is 30 cm. The length of the greatest side of the triangle in cm is

(a) 6

(b) 10

(c) 14

(d) 16

**Ques. **If each side of a square is increased by 25%, find the percentage change in its area.

(a) 47.25%

(b) 66.25%

(c) 62.25%

(d) 56.25%

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**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

**Ques. **The perimeters of two squares are 40 cm and 32 cm. Find the perimeter of a third square whose area is equal to the difference of the areas of the two squares.

(a) 32 cm

(b) 28 cm

(c) 24 cm

(d) 20 cm

**Ques. **How many triangles can be drawn by means of 9 non-collinear points

(a) 84

(b) 72

(c) 144

(d) 126

**Ques. **Find the area of a right-angled triangle whose base is 12 cm and hypotenuse is 13cm.

(a) 30 cm2

(b) 25 cm2

(c) 35 cm2

(d) 20 cm2

**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}

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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}

**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}

**Ques. **A room 5m 55cm long and 3m 74 cm broad is to be paved with square tiles. Find the least number of square tiles required to cover the floor.

(a) 173

(b) 174

(c) 175

(d) 176

**Ques. **If the diagonal of a rectangle is 17 cm long and its perimeter is 46 cm, find the area of the rectangle.

(a) 100 cm2.

(b) 120 cm2.

(c) 80 cm2.

(d) 150 cm2.

**Question:** The number of triangles that can be formed by 5 points in a line and 3 points on a parallel line, is

(a) ^{8}C_{3}

(b) ^{8}C_{3} – ^{5}C_{3}

(c) ^{8}C_{3} – ^{5}C_{3} – 1

(d) none of these

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**Ques. **Find the length of a rope by which a cow must be tethered in order that it may be able to graze an area of 9856 sq. metres.

(a) 58 m

(b) 48 m

(c) 56 m

(d) 52 m

**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

**Ques. **If the length of a certain rectangle is decreased by 4 cms and the width is increased by 3 cms, a square with the same area as the original rectangle would result. Find the perimeter of the original rectangle.

(a) 50 cms

(b) 51 cms

(c) 52 cms

(d) 53 cms

**Ques. **The difference between two parallel sides of a trapezium is 4 cms. perpendicular distance between them is 19 cms. If the area of the trapezium is 475 find the lengths of the parallel sides*.*

(a) 26 cms, 25 cms

(b) 27 cms, 23 cms

(c) 28 cms, 25 cms

(d) 25 cms, 22 cms

**Ques. **In two triangles, the ratio of the areas is 4 : 3 and the ratio of their heights is 3 : 4. Find the ratio of their bases.

(a) 6 : 7

(b) 16 : 9

(c) 4 : 5

(d) 16 : 15

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**Ques. **The length of a rectangle is twice its breadth. If its length is decreased by 5 cm and breadth is increased by 5 cm, the area of the rectangle is increased by 75 sq. cm. Find the length of the rectangle.

(a) 15.5 cm

(b) 17 cm

(c) 22 cm

(d) 20 cm

**Ques. **The diagonals of two squares are in the ratio of 2 : 5. Find the ratio of their areas.

(a) 3 : 4

(b) 4 : 5

(c) 4 : 25

(d) 6 : 11

**Ques. **Find the length of the altitude of an equilateral triangle of side 3√3 cm.

(a) 5.5 cms

(b) 6.5 cms

(c) 4.5 cms

(d) 7.5 cms

**Ques. **In measuring the sides of a rectangle, one side is taken 5% in excess, and the other 4% in deficit. Find the error percent in the area calculated from these measurements*.*

(a) 0.8%

(b) 1.8%

(c) 1.2%

(d) 1.0%

**Ques. **Find the area of a triangle whose sides measure 13 cm, 14 cm and 15 cm.

(a) 78 cm2

(b) 86 cm2

(c) 84 cm2

(d) 82 cm2

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**Ques. **The base of a parallelogram is twice its height. If the area of the parallelogram

is 72 sq. cm, find its height.

(a) 8 cms

(b) 6 cms

(c) 9 cms

(d) 5 cms

**Ques. **A rectangular grassy plot 110 m. by 65 m has a gravel path 2.5 m wide all round it on the inside. Find the cost of gravelling the path at 80 cents per sq. metre.

(a) $670

(b) $680

(c) $640

(d) $650

**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) - 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).