Question

A Field in the shape of a Trapezium whose parallel side area 25m and 10m. Non parallel sides are 14m and 13m. Find the area of the field

Answer 1

ANSWER:-

Given:

A field in the shape of a trapezium whose parallel side are 25m & 10m. The non-parallel sides are 14m & 13m.

To find:

Find the area of the field.

Solution:

Let the given field be in the shape of a trapezium ABCD in which

AB=25m,

CD=10m,

BC=13m,

AD=14m.

From D, draw DE||BC meeting AB at E.

& Draw DF AB.

Therefore,

DE= BC = 13m

AE=AB - EB = AB-DC

=) 25 - 10 = 15m.

So,

In ∆AED,

Using Heron's Formula:

⚫A= 14m

⚫B= 13m

⚫C= 15m

Therefore,

$s = \frac{A + B + C}{2} \\ \\ = > \frac{14 + 13 + 15}{2} \\ \\ = > \frac{42}{2} \\ \\ = > 21m$

Therefore,

Area of ∆AED,

$A = \sqrt{s(s - a)(s - b)(s - c)} \\ \\ = > \sqrt{21(21 - 14)(21 - 13)(21 - 15)} \\ \\ = > \sqrt{21(7)(8)(6)} \\ \\ = > \sqrt{3 \times 7 \times 7 \times 2 \times 4 \times 2 \times 3} \\ \\ = > 7 \times 3 \times 2 \times 2 \\ \\ = > 84 {m}^{2} \\ \\ = > \frac{1}{2} \times AE \times DF = 84 \\ \\ = > \frac{1}{2} \times 15 \times DF = 84 \\ \\ = > DF = \frac{84 \times 2}{15} \\ \\ = > DF = \frac{56}{5} m = 11.2m$

Height of trapezium is 11.2m.

Therefore,

We know that area of ||gm=Base× height

=) EB × DF

=) 10× 11.2

=) 112m²

Now,

Area of field:

Area of ∆AED + Area of ||gm EBCD

=) 84m² + 112m²

=) 196m².

Hope it helps ☺️

Answer 2

Answer :-

Area of the field is 196 m².

Explanation :-

Given :

• Field is in the shape of a trapezium

• Parallel sides - 25 m and 10 m

• Non parallel sides - 14 m and 13 m

To find :

Area of the field

Construction :-

• Draw a line segment DE from D to E such that DE || BC

• Draw perpendicular DF to AB

For figure refer the attachment

Solution :-

DC || AB so DC || EB (Given)

DE || CB (By construction)

So DECB is a parallelogram

DC = EB = 10 m

BC = DE = 13 m

Consider Δ AED

Lengths of the sides of the Δ AED are

• a = AB = 14 m

• b = DE = 13 m

• c = AE = AB - EB = 25 - 10 = 15 m

Finding area of the triangle by heron's formula

Semi peimeter (s) = (a + b + c)/2

s = (14 + 13 + 15)/2

s = 42/2

s = 21

Area of the Δ AED = √{ s(s - a)(s - b)(s - c)}

= √{21(21 - 14)(21 - 13)(21 - 15)}

= √{21(7)(8)(6)}

= √(7 * 3 * 7 * 2 * 2 * 2 * 2 * 3)

= √(7² * 3² * 2² * 2²)

= √{(7 * 3 * 2 * 2)²}

= 7 * 3 * 2 * 2

= 21 * 4

= 84 m²

Area of the Δ AED = 84 m²

Finding height of the parallelogram DEBC

In Δ AED

Base = AE = 15 m

Height = DF = 'x' m

Also Area of the Δ AED = (1/2) * Base * Height

⇒ 84 = (1/2) * 15 * x

⇒ 84 * 2 = 15x

⇒ 168 = 15x

⇒ 168/15 = x

⇒ 11.2 = x

⇒ x = 11.2

Height of the Δ AED = DF = 11.2 m

DF is also height for parallelogram DEBC

So Height of the parallelogram DEBC = 11.2 m

Finding the area of || gm DEBC

In || gm DEBC

Base = EB = 10 m

Height = DF = 11.2 m

Area of the || gm = Base * Height = 10 * 11.2 = 112 m²

Finding the area of the field

Area of the field trapezium ABCD = Area of the || gm DEBC + Area of the Δ AED

= 112 + 84

= 196 m²

Therefore the area of the field is 196 m².