Trapezium questions and answers are available here to assist students in learning more about trapezium properties and how to solve problems related to them. As we know, the trapezium is one of the quadrilaterals we learn in our classes, which has exactly one pair of parallel sides. In this article, you will get the solved and practice questions on trapezium, which will enhance your geometrical skills.
What is Trapezium?
A trapezium is a two-dimensional figure in which two sides are parallel to each other, and one pair of sides are non-parallel. In other words, a trapezium is a quadrilateral with one pair of parallel sides. The non-parallel sides are called legs.
Suppose a, b, c, and d are the sides of a trapezium, such that a || c then,
Perimeter = a + b + c + d
Area = (½) × (a + c) × h
Here, h is the height or the distance between parallel sides.
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Trapezium Questions and Answers
1. Find the perimeter of a trapezium with sides 4 cm, 6 cm, 7 cm and 9 cm.
Solution:
Let the sides of a trapezium be:
a = 4 cm, b = 6 cm, c = 7 cm and d = 9 cm
Perimeter of a trapezium = a + b + c + d
= 4 + 6 + 7 + 9
= 26 cm
Thus, the perimeter of the trapezium is 26 cm.
2. If the perimeter of a trapezium is 32 units, whose three sides are 3 units, 8 units and 10 units. Find the measure of the fourth side.
Solution:
Let the sides of a trapezium be:
a = 3 units, b = 8 units, c = 10 units and d = x units
Perimeter of a trapezium = a + b + c + d
3 + 8 + 10 + x = 32 units (given)
21 + x = 32
x = 32 – 21 = 11
Therefore, the fourth side of the trapezium measures 11 units.
3. What is isosceles trapezium?
Solution:
In geometry, an isosceles trapezium is a trapezium in which non-parallel sides are equal in length. The below figure depicts the shape of an isosceles trapezium.
4. The perimeter of a trapezium is 58 cm, and the sum of its non-parallel sides is 20 cm. If its area is 152 cm2, then what is the distance between the parallel sides?
Solution:
Given,
Sum of non-parallel sides of a trapezium = 20 cm
Perimeter = 58 cm
Sum of parallel sides of the trapezium = Perimeter – Sum of non-parallel sides
= 58 – 20
= 38 cm
Area of trapezium = (½) × Sum of parallel sides × Distance between parallel sides
152 = (½) × 38 × Distance between parallel sides
⇒ Distance between parallel sides × 19 = 152
⇒ Distance between parallel sides = 152/19 = 8
Therefore, the distance between parallel sides is 8 cm.
5. If the perimeter of a trapezium is 52 cm, its non-parallel sides are equal to 10 cm each, and its altitude is 8 cm, find the trapezium area.
Solution:
Given,
Perimeter of a trapezium = 52 cm
Measure of non-parallel sides is 10 cm each.
Sum of parallel sides = Perimeter of trapezium – Sum of non-parallel sides
= 52 – (10 + 10)
= 52 – 20
= 32 cm
Also, given that the height (distance between the parallel sides) = 8 cm
Area of the trapezium = (½) × Sum of parallel sides × Distance between parallel sides
= (½) × 32 × 8
= 128 cm2
6. If the area of a trapezium is 28 cm2 and one of its parallel sides is 6 cm, find the other parallel side if its altitude is 4 cm.
Solution:
Given,
Altitude or distance between parallel sides = 4 cm
One of the parallel sides = 6 cm
Let x be the other parallel side of the trapezium.
Area of the trapezium = (½) × Sum of parallel sides × Distance between parallel sides
(½) × (6 + x) × 4 = 28 cm2 (given)
6 + x = 28/2
6 + x = 14
x = 14 – 6 = 8
Hence, the other parallel sides of the given trapezium are 8 cm.
7. The area of a trapezium is 384 cm2. Its parallel sides are in the ratio 3 : 5, and the perpendicular distance between them is 12 cm. Find the length of each one of the parallel sides.
Solution:
Given,
The ratio of parallel sides of a trapezium = 3 : 5
Let 3x and 5x be the length of parallel sides (in cm).
Also, given that the perpendicular distance between parallel sides = 12 cm
Area of trapezium = 384 cm2
Area of the trapezium = (½) × Sum of parallel sides × Distance between parallel sides
(½) × (3x + 5x) × 12 = 384
8x = 384/6
8x = 64
x = 64/8 = 8
So, 3x = 3(8) = 24 cm
5x = 5(8) = 40 cm
Therefore, the lengths of parallel sides are 24 cm and 40 cm.
8. The parallel sides of a trapezium are 25 m and 10 m and the non-parallel sides are 14 m and 13 m. Calculate the distance between the parallel sides and the area of the trapezium.
Solution:
Let ABCD be the trapezium, such that AB and CD are parallel sides.
That means, AB = 10 m, BC = 14 m, CD = 25 m and DA = 13 m.
Draw BE parallel to the DA, such that ABED is a parallelogram.
Also, draw BF perpendicular to CE or CD.
Thus, AB = DE = 10 m
BE = DA = 13 m
Also, CE = CD – DE = (25 – 10) m = 15 m
Consider triangle BEC.
Semiperimeter (s) = (BE + EC + BC)/2
= (13 + 15 + 14)/2
= 42/2
= 21 m
Area of ΔBEC = √[s(s – a)(s – b)(s – c)] {by Heron’s formula}
= √[21(21 – 3)(21 – 14)(21 – 15)]
= √(21 × 18 × 7 × 6)
= 84 m2
Also, area of triangle BEC = (½) × CE × BF
(½) × 15 × BF = 84
BF = (84 × 2)/15
= 11.2 m
That means the distance between parallel sides of the trapezium = 11.2 m
Now, are of the trapezium = (½) × Sum of parallel sides × Distance between parallel sides
= (½) × (10 + 25) × 11.2
= 196 m2
9. ABCD is a trapezium, such that AD and BC are the parallel sides. If ∠B – ∠A = 40° and ∠C – ∠D = 20°, then find the value of ∠B + ∠C.
Solution:
Given,
ABCD is a trapezium, such that AD and BC are the parallel sides.
∠B – ∠A = 40°….(1)
∠C – ∠D = 20°….(2)
∠B – ∠A + ∠C – ∠D = 40° + 20°
∠B + ∠C = 60° + ∠A + ∠D
∠A + ∠D = ∠B + ∠C – 60°….(3)
As we know, sum of all interior angles of a trapezium = 360°
∠A + ∠B + ∠C + ∠D = 360°
∠B + ∠C + ∠B + ∠C – 60° = 360°
2(∠B + ∠C) = 360° + 60°
∠B + ∠C = 420°/2
∠B + ∠C = 210°
10. In a trapezium PQRS, PQ || RS, ∠P : ∠S = 3 : 2 and ∠Q : ∠R = 4 : 5. Find the angles of the trapezium.
Solution:
Given,
∠P : ∠S = 3 : 2
∠Q : ∠R = 4 : 5
Let 3x and 2x be the measures of angles P and S.
And let 4x and 5x be the measures of angle Q and R.
As we know, the sum of adjacent angles at the corners of non-parallel sides of a trapezium = 360°
∠P + ∠S = 180°
3x + 2x = 360°
5x = 180°
x = 180°/5 = 36°
So, 3x = 3(36°) = 108°
2x = 2(36°) = 72°
Similarly,
4x + 5x = 180°
9x = 180°
x = 180°/9 = 20°
So, 4x = 4(20°) = 80°
5x = 5(20°) = 100°
Therefore, the angles of the trapezium are ∠P = 108°, ∠Q = 80, ∠R = 100 and ∠S = 72.
Practice Questions on Trapezium
- Find the area of a trapezium whose parallel sides are 25 units, 13 units and the other sides are 15 units each.
- The area of a trapezium is 1586 cm2, and the distance between the parallel sides is 26 cm. If one of the parallel sides is 38 cm, find the other.
- In the given parallelogram YOUR, ∠RUO = 120° and OY is extended to point S, such that ∠SRY = 50°. Find ∠YSR.
- Construct a trapezium ABCD in which AB||DC, ∠A = 105°, AD = 3 cm, AB = 4 cm and CD = 8 cm.
- The area of a trapezium is 91 cm2, and its height is 7 cm. If one of the parallel sides is longer than the other by 8 cm, find the two parallel sides.
