How to find the parallel side of an isosceles trapezoid?

Mensuration entails the processes of measurements and all calculations pertaining to various geometrical shapes, occurring in mathematical theory as well as our daily lives. The study of all geometrical shapes falls under the scope of mensuration. 

Trapezium

Such a convex quadrilateral which has exactly one pair of parallel sides is called a trapezium or trapezoid. The trapezium is a 2-D shape that has the appearance of a table. Like all other quadrilaterals, the sum of all the interior angles of a trapezium is also 360°. The following figure shows a trapezium ABCD, with AB and CD as its parallel sides and AD and BC as its non-parallel sides. 

Isosceles Trapezoid

An isosceles trapezoid is a trapezoid that has exactly one pair of parallel sides and the non-parallel sides are equal in length. The base angles are of equal measure as well.

The figure above depicts an isosceles trapezoid ABCD, with AB and CD being the base sides and AD and BC being the non-parallel sides equal in length. The base angles: ∠B = ∠A and ∠C = ∠D.

Properties of an isosceles trapezoid

• An isosceles trapezoid is the one whose base sides are parallel.
• The sides other than the base sides are non-parallel and equal in length.
• The diagonals of an isosceles trapezoid are equal in length.
• The base angles of an isosceles trapezoid are equal in measure.

Area of isosceles trapezoid

The area of an isosceles trapezoid is calculated using the following formula,

A = (Height × sum of parallel sides)/2

How to find the parallel side of an isosceles trapezoid?

Answer:

Given the ratio of the parallel sides of an isosceles trapezium and its area, the lengths of each side can be calculated with ease.

Example: Say the ratio of parallel sides of an isosceles trapezoid is 2:1. The area is 90 unit² and height is 10 units.

Let the length of the shorter parallel side be ‘x’ units. 

A = (Height × sum of parallel sides)/2

⇒ 90 = 10(x + 2x)/ 2

⇒ 18 = 3x

⇒ x = 6 units

Thus, the length of the shorter side is 6 units and that of the longer parallel side is 12 units.

Similar Problems

Question 1: Find the area of an isosceles trapezoid whose perimeter is 100 cm, height 10 cm, and non-parallel side is 20 cm each.

Solution:

Perimeter = Sum of all sides = 100 cm.

Let the sum of parallel sides be  a + b.

⇒ a + b + 20 + 20 = 100

⇒ a + b = 60 cm

Now, Area = h(a + b)/ 2

= 10(60)/ 2

⇒ A = 300 cm²

Question 2: Find the lengths of parallel sides of an isosceles trapezoid given their ratio is 1:3, height is 5 units and the area is 100 units².

Solution:

Let the length of the shorter parallel side be x units.

A = (Height × sum of parallel sides)/2

⇒ 100 = 5(x + 3x)/ 2

⇒ 40 = 4x

⇒ x = 10 units

Thus, the length of the shorter side is 10 units and that of the longer parallel side is 30 units.

Question 3: Find the lengths of parallel sides of an isosceles trapezoid given their ratio is 1:4, height is 5 units and the area is 100 units².

Solution:

Let the length of the shorter parallel side be x units.

A = (Height × sum of parallel sides)/2

⇒ 100 = 5(x + 4x)/ 2

⇒ 40 = 5x

⇒ x = 8 units

Thus, the length of the shorter side is 8 units and that of the longer parallel side is 32 units.

Question 4: Find the lengths of parallel sides of an isosceles trapezoid given their ratio is 1:4, height is 5 units and the area is 50 units².

Solution:

Let the length of the shorter parallel side be x units.

A = (Height × sum of parallel sides)/2

⇒ 50 = 5(x + 4x)/ 2

⇒ 20 = 5x

⇒ x = 4 units

Thus, the length of the shorter side is 4 units and that of the longer parallel side is 16 units.

Question 5: Find the lengths of parallel sides of an isosceles trapezoid given their ratio is 1:3, height is 2 units and the area is 100 units².

Solution:

Let the length of the shorter parallel side be x units.

A = (Height × sum of parallel sides)/2

⇒ 100 = 2(x + 3x)/ 2

⇒ 100 = 4x

⇒ x = 25 units

Thus, the length of the shorter side is 25 units and that of the longer parallel side is 75 units.