b. Calculate the area of the shaded region
Answers
Required Answer :
The area of shaded region = 54 cm²
Given :
- OB = 5 cm
- AO = 12 cm
- BC = 14 cm
- AC = 15 cm
To find :
- The area of shaded region
Solution :
Firstly, we will calculate the length of AB, by using the pythagoras theorem.
In ∆AOB
By pythagoras theorem,
⇒ H² = P² + B²
where,
- H = Hypotenuse
- P = Perpendicular
- B = Base
we have,
- Hypotenuse (AB) = ?
- Perpendicular (OB) = 5 cm
- Base (AO) = 12 cm
Substituting the values :
⇒ AB² = (5)² + (12)²
⇒ AB² = 25 + 144
⇒ AB² = 169
⇒ Taking square root on both the sides :
⇒ AB = √169
⇒ AB = √(13 × 13)
⇒ AB = ± 13 Reject - ve
⇒ AB = 13 cm
Now let's calculate the area of ∆ABC by using the Heron's formula.
- Semi perimeter = (a + b + c) ÷ 2
where,
- a, b and c are the three sides of the triangle
Substituting the given values :
⇒ Semi perimeter = (13 + 15 + 14) ÷ 2
⇒ Semi perimeter = 42 ÷ 2
⇒ Semi perimeter = 21 cm
- Heron's formula = √s(s - a)(s - b)(s - c)
Substituting the given values :
⇒ Area = √21(21 - 13)(21 - 15)(21 - 14)
⇒ Area = √21(8)(6)(7)
⇒ Area = √(7 × 3 × 2 × 2 × 2 × 3 × 2 × 7)
⇒ Area = 7 × 3 × 2 × 2
⇒ Area = 84
- Area of ∆ABC = 84 cm²
Now, calculating the area of ∆AOB, by using the following formula :
- Area of triangle = ½ × b × h
where,
- b denotes the base
- h denotes the height
we have,
- b = 12 cm
- h = 5 cm
Substituting the given values :
⇒ Area = ½ × 12 × 5
⇒ Area = 6 × 5
⇒ Area = 30
- Area of ∆AOB = 30 cm²
To calculate the area of shaded region, subtract the area of ∆AOB from the area of ∆ABC. The resultant value will be the area of shaded region.
⇒ Area of shaded region = Area of ∆ABC - Area of ∆AOB
⇒ Area of shaded region = 84 - 30
⇒ Area of shaded region = 54
The area of shaded region = 54 cm²