Question

find the area of the quadrilateral where sides measure 9 cm, 40 cm, 28 cm and 15 cm and in which the angles first two sides is 90 degree. use the heron's formula for this question.​

Answer 1

Given:-

• Sides of quadrilateral are 9 cm , 40 cm, 28 cm and 15 cm.
• Angle between first two side is 90°.

To find:-

• Area of quadrilateral.

SolutiOn:-

Let the quadrilateral be ABCD.

And Sides be 9 cm be AB, 40 cm be BC , 28 cm be DC and 15 cm be AD.

Construct a diagonal AC.

Two triangles will form 1st and 2nd triangle

______________________

• We need area of quadrilateral. So, Area of quadrilateral will be sum of of areas of triangle 1st and triangle 2nd.

________________

In triangle 1st,

We know that,

Pythagoras theorem is,

◆ (Hypotenuse)² = (Perpendicular)² + (Base)² ◆

Hypotenuse = AC = ?

Perpendicular = AB = 9 cm.

Base = BC = 40 cm

Put the values,

⇒ (AC)² = (9)² + (40)²

⇒ (AC)² = 1681

⇒ AC = √1681

⇒ AC = 41 -------(i)

We know that,

Area of triangle = 1/2 × base × height

Height = AB = 9 cm

Base = BC = 40 cm

Put the values

⇒ 1/2 × 9 × 40

⇒ 9 × 20

⇒ 180

Area of 1st triangle is 180 cm².

________________

In triangle 2nd

Height is not given.So we use Heron's formula that is,

$\sf \sqrt{s(s - a)(s - c)(s - c)}$

In which,

• S is semi-perimeter
• a , b and c are sides of triangle

a = AC = 41 [From equation (i)]

b = CD = 28 cm

c = AD = 15 cm.

So,

Semi-perimeter = Perimeter of triangle/2

• Perimeter of triangle is sum of all sides if triangle.

= 41 + 28 + 15/2

= 84/2

= 42

Semi-perimeter of triangle is 42 cm.

Area of triangle

= $\sf \sqrt{42\times (42 - 15)(42 - 28)(42 - 41)}$

= $\sf \sqrt{42 \times 27 \times 14 \times 1}$

=$\sf \sqrt{2 \times 3 \times 7 \times 3 \times 3 \times 3 \times 7 \times 2}$

=$\sf 2 \times 3 \times 7 \times 3$

= $\sf 126$

Area of 2nd triangle is 126 cm².

____________________

Area of ABCD = Area of 1st triangle + Area of 2nd triangle.

⇒ 180 + 126

⇒ 306 cm

Area of ABCD is 306 cm².

As we have taken ABCD to be quadrilateral.

___________________

Therefore,

Area of quadrilateral is 306 cm².