Question

Look at the measures shown in the adjacent figure and find the area of ☐ PQRS.​

Answer 1

Given:

✰ PS = 36 m

✰ SR = 15 m

✰ RQ = 25 m

✰ PQ = 56 m

To find:

✠ The area of ☐ PQRS.

Solution:

❖ Let's understand the concept first! First of all we will find the area of ∆PSR and then the are of ∆PQR. To find the area of ∆PSR, we will simply use formula of triangle, but to find the area of ∆PQR, first we will calculate it's side PR, then using Heron's formula we will calculate it's area. After that we will add both the area of ∆PSR and the area of ∆PQR to get the area of whole figure i.e, of ☐ PQRS.

✭Area of triangle = 1/2 × base × height✭

• Area of ∆PSR = 1/2 × b × h

• Area of ∆PSR = 1/2 × 15 × 36

• Area of ∆PSR = 1/2 × 15 × 36

• Area of ∆PSR = 1 × 15 × 18

• Area of ∆PSR = 270 m²

Now,

By using Pythagoras theorem,

⇾H² = B² + P²

⇾PR² = SR² + PS²

⇾PR² = 15² + 36²

⇾PR² = 225 + 1296

⇾PR² = 1521

⇾PR = √1521

⇾PR = 39

Using Heron's formula,

✭ Semi-perimeter ( S ) = (a+b+c)/2 ✭

• S = (39 + 56 + 25)/2
• S = (95 + 25)/2
• S = 120/2
• S = 60

✭ Area of triangle = √s( s - a ) ( s - b ) ( s - c ) ✭

• Area of ∆PQR = √s( s - a ) ( s - b ) ( s - c )

• Area of ∆PQR = √60( 60 - 39 ) ( 60 - 56 ) ( 60 - 25 )

• Area of ∆PQR = √60( 21 ) ( 4 ) ( 35 )

• Area of ∆PQR = √(60 × 21 × 4 × 35)

• Area of ∆PQR = √(3 × 2 × 2 × 5 × 3 × 7 × 2 × 2 × 7 × 5 )

• Area of ∆PQR = √( 3 × 3 × 2 × 2 × 5 × 5 × 7 × 7 × 2 × 2)

• Area of ∆PQR = √( 3 × 3 × 2 × 2 × 5 × 5 × 7 × 7 × 2 × 2)

• Area of ∆PQR = 3 × 2 × 5 × 7 × 2

• Area of ∆PQR = 420 m²

➤ Area of ☐ PQRS = Area of ∆PSR + Area of ∆PQR

➤ Area of ☐ PQRS = ( 270 + 420 ) m²

➤ Area of ☐ PQRS = 690 m²

∴ The area of ☐ PQRS = 690 m²

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