CBSE Class 9 Mathematics Areas of Parallelograms and Triangles MCQs Set 03

Multiple Choice Questions (MCQs) for Class 9 Mathematics: Chapter 09 Areas Of Parallelograms And Triangles

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Practice Chapter 09 Areas Of Parallelograms And Triangles MCQs for Class 9 Mathematics

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Multiple Choice Questions

Question 1: Two opposite angles of a parallelogram are \( (3x-2)^\circ \) and \( (50-x)^\circ \). The angles of the parallelogram are
(a) \( 37^\circ,\ 143^\circ,\ 37^\circ,\ 143^\circ \)
(b) \( 40^\circ,\ 140^\circ,\ 40^\circ,\ 140^\circ \)
(c) \( 37^\circ,\ 37^\circ,\ 143^\circ,\ 37^\circ \)
(d) \( 13^\circ,\ 167^\circ,\ 13^\circ,\ 167^\circ \)
Show Answer & Explanation

Answer: (a) \( 37^\circ,\ 143^\circ,\ 37^\circ,\ 143^\circ \)

Explanation:
1. Opposite angles are equal: \( 3x-2=50-x \), so \( x=13 \).
2. The angle is \( 3(13)-2=37^\circ \) and the adjacent angle is \( 180-37=143^\circ \).

Teacher's Note:
1. Opposite angles of a parallelogram are equal.

Question 2: One angle of a parallelogram is two-thirds of its adjacent angle. The angles of the parallelogram are
(a) \( 60^\circ,\ 120^\circ,\ 60^\circ,\ 120^\circ \)
(b) \( 72^\circ,\ 108^\circ,\ 72^\circ,\ 108^\circ \)
(c) \( 54^\circ,\ 126^\circ,\ 54^\circ,\ 126^\circ \)
(d) \( 80^\circ,\ 100^\circ,\ 80^\circ,\ 100^\circ \)
Show Answer & Explanation

Answer: (b) \( 72^\circ,\ 108^\circ,\ 72^\circ,\ 108^\circ \)

Explanation:
1. Let the larger adjacent angle be \( x \): \( x+\frac23x=180 \), so \( x=108 \).
2. The other is \( 72^\circ \).

Teacher's Note:
1. Adjacent angles are supplementary.

Question 3: One angle of a parallelogram is \( 24^\circ \) less than twice the smallest angle. Its largest angle is
(a) \( 68^\circ \)
(b) \( 124^\circ \)
(c) \( 112^\circ \)
(d) \( 136^\circ \)
Show Answer & Explanation

Answer: (c) \( 112^\circ \)

Explanation:
1. Let the smallest angle be \( x \), so the adjacent angle is \( 180-x \).
2. \( 180-x=2x-24 \) gives \( x=68 \), so the largest angle is \( 180-68=112^\circ \).

Teacher's Note:
1. Adjacent angles are supplementary.

Question 4: The perimeter of a parallelogram is \( 22 \) cm and the longer side is \( 6.5 \) cm. The shorter side is
(a) \( 5 cm \)
(b) \( 9 cm \)
(c) \( 4.45 cm \)
(d) \( 4.5 cm \)
Show Answer & Explanation

Answer: (d) \( 4.5 cm \)

Explanation:
1. Semi-perimeter \( =11 \) cm, so the shorter side \( =11-6.5=4.5 \) cm.

Teacher's Note:
1. Semi-perimeter is the sum of adjacent sides.

Question 5: In parallelogram \( ABCD \), \( \angle D=135^\circ \). Then \( \angle A \) and \( \angle B \) are
(a) \( 45^\circ\text{ and }135^\circ \)
(b) \( 135^\circ\text{ and }45^\circ \)
(c) \( 45^\circ\text{ and }45^\circ \)
(d) \( 90^\circ\text{ and }135^\circ \)
Show Answer & Explanation

Answer: (a) \( 45^\circ\text{ and }135^\circ \)

Explanation:
1. \( \angle A \) is adjacent to \( \angle D \): \( \angle A=180-135=45^\circ \).
2. \( \angle B \) is opposite to \( \angle D \): \( \angle B=135^\circ \).

Teacher's Note:
1. Adjacent supplementary, opposite equal.

Question 6: In parallelogram \( ABCD \), \( \angle DAB=75^\circ \) and \( \angle DBC=60^\circ \). Then \( \angle BDC \) is
(a) \( 60^\circ \)
(b) \( 45^\circ \)
(c) \( 75^\circ \)
(d) \( 30^\circ \)
Show Answer & Explanation

Answer: (b) \( 45^\circ \)

Explanation:
1. \( AD\parallel BC \) gives \( \angle ADB=\angle DBC=60^\circ \).
2. In \( \triangle ABD \): \( \angle ABD=180-75-60=45^\circ \).
3. \( AB\parallel DC \) gives \( \angle BDC=\angle ABD=45^\circ \).

Teacher's Note:
1. Use alternate angles twice.

Question 7: The ratio of the areas of two parallelograms on equal bases and between the same parallels is
(a) \( 1 : 2 \)
(b) \( 2 : 1 \)
(c) \( 1 : 1 \)
(d) \( depends on the angles \)
Show Answer & Explanation

Answer: (c) \( 1 : 1 \)

Explanation:
1. Parallelograms on equal bases and between the same parallels have equal areas (same base and same height).

Teacher's Note:
1. Area = base times height.

Question 8: A rectangle and a parallelogram have equal areas, the same base and lie on the same side of it. The ratio (perimeter of rectangle) : (perimeter of parallelogram) is
(a) greater than 1
(b) equal to 1
(c) cannot be compared
(d) less than 1
Show Answer & Explanation

Answer: (d) less than 1

Explanation:
1. They are between the same parallels, so the rectangle's side is the perpendicular distance, which is shorter than the parallelogram's slanting side.
2. With the same base, the rectangle has the smaller perimeter.

Teacher's Note:
1. The perpendicular is the shortest segment.

Question 9: \( E,F \) are mid-points of \( BC \) and \( AD \) of parallelogram \( ABCD \) and \( G \) is any point on \( EF \). Then \( \text{ar}(\triangle GAB) \) is
(a) \( \frac14\text{ar}(ABCD) \)
(b) \( \frac12\text{ar}(ABCD) \)
(c) \( \frac13\text{ar}(ABCD) \)
(d) \( \frac18\text{ar}(ABCD) \)
Show Answer & Explanation

Answer: (a) \( \frac14\text{ar}(ABCD) \)

Explanation:
1. \( EF\parallel AB \), so the distance of \( G \) from \( AB \) is half the height of the parallelogram.
2. \( \text{ar}(GAB)=\frac12\cdot AB\cdot\frac h2=\frac14\cdot AB\cdot h=\frac14\text{ar}(ABCD) \).

Teacher's Note:
1. G is at half the height.

Question 10: The altitude of a parallelogram is twice its base and its area is \( 1250 \) cm\( ^2 \). The base and altitude are
(a) \( 50 cm and 25 cm \)
(b) \( 25 cm and 50 cm \)
(c) \( 25 cm and 25 cm \)
(d) \( 12.5 cm and 25 cm \)
Show Answer & Explanation

Answer: (b) \( 25 cm and 50 cm \)

Explanation:
1. \( b\times2b=1250 \), so \( b^2=625 \) and \( b=25 \).
2. The altitude is \( 50 \) cm.

Teacher's Note:
1. Area = base times altitude.

Question 11: \( E \) is a point on side \( DC \) of parallelogram \( ABCD \) of area \( 30 \) cm\( ^2 \). Then \( \text{ar}(ADE)+\text{ar}(BCE) \) is
(a) \( 30\text{ cm}^2 \)
(b) \( 10\text{ cm}^2 \)
(c) \( 15\text{ cm}^2 \)
(d) \( 20\text{ cm}^2 \)
Show Answer & Explanation

Answer: (c) \( 15\text{ cm}^2 \)

Explanation:
1. \( \text{ar}(ABE)=\frac12\text{ar}(ABCD)=15 \) (same base \( AB \), between \( AB\parallel DC \)).
2. The rest of the parallelogram is \( \text{ar}(ADE)+\text{ar}(BCE)=30-15=15 \) cm\( ^2 \).

Teacher's Note:
1. Triangle ABE is half the parallelogram.

Question 12: \( E \) is on \( DC \) and \( F \) is on \( AD \) of parallelogram \( ABCD \). If \( \text{ar}(\triangle AEB)=16 \) cm\( ^2 \), then \( \text{ar}(\triangle BFC) \) is
(a) \( 8\text{ cm}^2 \)
(b) \( 32\text{ cm}^2 \)
(c) \( 24\text{ cm}^2 \)
(d) \( 16\text{ cm}^2 \)
Show Answer & Explanation

Answer: (d) \( 16\text{ cm}^2 \)

Explanation:
1. \( \text{ar}(AEB)=\frac12\text{ar}(ABCD) \) and \( \text{ar}(BFC)=\frac12\text{ar}(ABCD) \) (base \( BC \), between \( BC\parallel AD \)).
2. So both are \( 16 \) cm\( ^2 \).

Teacher's Note:
1. Each triangle is half the parallelogram.

Question 13: The ratio of the altitude to the area of a parallelogram is 2 : 11. The length of the base is
(a) \( 5.5 units \)
(b) \( 2.5 units \)
(c) \( 11 units \)
(d) \( 22 units \)
Show Answer & Explanation

Answer: (a) \( 5.5 units \)

Explanation:
1. Altitude \( =2k \), area \( =11k \), so base \( =\frac{11k}{2k}=5.5 \).

Teacher's Note:
1. Base = area divided by altitude.

Question 14: Two adjacent sides of a parallelogram are \( 24 \) cm and \( 18 \) cm. If the distance between the longer sides is \( 12 \) cm, the distance between the shorter sides is
(a) \( 9 cm \)
(b) \( 16 cm \)
(c) \( 18 cm \)
(d) \( 12 cm \)
Show Answer & Explanation

Answer: (b) \( 16 cm \)

Explanation:
1. Area \( =24\times12=288 \) cm\( ^2 \).
2. Distance between the shorter sides \( =\frac{288}{18}=16 \) cm.

Teacher's Note:
1. Area is the same for either base.

Question 15: \( ABCD \) is a trapezium with \( AB=a \), \( DC=b \), \( AB\parallel DC \). \( E,F \) are the mid-points of the non-parallel sides \( AD,BC \). The ratio \( \text{ar}(ABFE):\text{ar}(EFCD) \) is
(a) \( a:b \)
(b) \( (a+b):(a-b) \)
(c) \( (3a+b):(a+3b) \)
(d) \( (2a+b):(a+2b) \)
Show Answer & Explanation

Answer: (c) \( (3a+b):(a+3b) \)

Explanation:
1. \( EF=\frac{a+b}2 \) and the two trapeziums have the same height \( \frac h2 \).
2. \( \text{ar}(ABFE)=\frac12\left(a+\frac{a+b}2\right)\frac h2 \) and \( \text{ar}(EFCD)=\frac12\left(\frac{a+b}2+b\right)\frac h2 \).
3. Ratio \( =(3a+b):(a+3b) \).

Teacher's Note:
1. Both pieces have the same height.

Question 16: The area of a triangle equals the area of a rectangle 18 cm by 12 cm. If the base of the triangle is 24 cm, its altitude is
(a) \( 9 cm \)
(b) \( 12 cm \)
(c) \( 36 cm \)
(d) \( 18 cm \)
Show Answer & Explanation

Answer: (d) \( 18 cm \)

Explanation:
1. Area \( =216 \) cm\( ^2 \).
2. \( \frac12\times24\times h=216 \), so \( h=18 \) cm.

Teacher's Note:
1. Half base times height.

Question 17: In a parallelogram, the diagonals are
(a) bisecting each other but not necessarily equal
(b) equal but not bisecting each other
(c) perpendicular but not bisecting each other
(d) always equal and perpendicular
Show Answer & Explanation

Answer: (a) bisecting each other but not necessarily equal

Explanation:
1. The diagonals of a parallelogram bisect each other; they are equal only in a rectangle and perpendicular only in a rhombus.

Teacher's Note:
1. Check with a rhombus and a rectangle.

Question 18: Which statement is true for every parallelogram?
(a) The diagonals are equal
(b) The diagonals bisect each other
(c) The diagonals meet at right angles
(d) The diagonals bisect the angles
Show Answer & Explanation

Answer: (b) The diagonals bisect each other

Explanation:
1. Only 'the diagonals bisect each other' holds for every parallelogram.

Teacher's Note:
1. Test with a general parallelogram.

Question 19: If \( ABCD \) is a parallelogram with adjacent angles \( A \) and \( B \) equal, then it is a
(a) rhombus
(b) trapezium
(c) rectangle
(d) kite
Show Answer & Explanation

Answer: (c) rectangle

Explanation:
1. \( \angle A+\angle B=180 \) and \( \angle A=\angle B \), so each is \( 90^\circ \); a parallelogram with a right angle is a rectangle.

Teacher's Note:
1. Adjacent angles add to 180.

Question 20: \( E,F \) are the centroids of triangles \( ABD \) and \( BCD \) in parallelogram \( ABCD \). Then
(a) \( EF = 2AE \)
(b) \( EF = \frac12AE \)
(c) \( EF = AC \)
(d) \( EF = AE \)
Show Answer & Explanation

Answer: (d) \( EF = AE \)

Explanation:
1. With \( O \) the centre, the centroids lie on \( AC \): \( AE=\frac23AO=\frac13AC \) and \( CF=\frac13AC \).
2. \( EF=AC-AE-CF=\frac13AC=AE \).

Teacher's Note:
1. Centroid divides the median in the ratio 2 : 1.

Question 21: The diagonal of a parallelogram divides it into two triangles which
(a) have equal areas (they are congruent)
(b) have unequal areas
(c) are always right-angled
(d) are always isosceles
Show Answer & Explanation

Answer: (a) have equal areas (they are congruent)

Explanation:
1. A diagonal divides a parallelogram into two congruent triangles.

Teacher's Note:
1. Diagonal gives two congruent triangles.

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Chapter 09 Areas Of Parallelograms And Triangles Objective Questions & Solutions for Class 9 Mathematics

Chapter MCQs with Answers for Class 9 Mathematics

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FAQs

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