find the area of triangle whose sides are 20 CM ,21 cm and 13 cm find the length of the altitude corresponding to the longest side
Answers
Step-by-step explanation:
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Required Answer :
The area of triangle = 126 cm²
The length of the altitude corresponding to the longest side = 12 cm
Given :
- Sides of a triangle :
- First side = 20 cm
- Second side = 21 cm
- Third side = 13 cm
To find :
- Area of the triangle
- Length of the altitude corresponding to the longest side
Solution :
To calculate the area of the triangle, we will use the Heron's formula. For that firstly, we will calculate the semi perimeter of the triangle.
Formula to calculate the semi perimeter :
- Semi perimeter = Perimeter ÷ 2
or
- Semi perimeter = (a + b + c) ÷ 2
where,
- a, b and c denotes the three sides of the triangle
we have,
- a = 20 cm
- b = 21 cm
- c = 13 cm
⇒ Semi perimeter = (20 + 21 + 13) ÷ 2
⇒ Semi perimeter = 54 ÷ 2
⇒ Semi perimeter = 27
- Semi perimeter = 27 cm
Using formula,
- Heron's formula = √s(s - a)(s - b)(s - c)
Substituting the given values :
⇒ Area = √27(27 - 20)(27 - 21)(27 - 13)
⇒ Area = √27(7)(6)(14)
⇒ Area = √15876
⇒ Area = √126 × 126)
⇒ Area = ± 126
As we know, area of triangle cannot be negative. So, the negative sign will get rejected.
⇒ Area = ± 126 Reject -ve
⇒ Area = 126
Therefore, the area of triangle = 126 cm²
The longest side of the triangle = 21 cm
Using formula,
- Area of triangle = ½ × b × h
where,
- b = base
- h = height
Substituting the given values :
⇒ 126 = ½ × 21 × h
⇒ 126 = 21/2 = h
⇒ 126 × 2/21 = h
⇒ 12 = h
Therefore, the length of the altitude corresponding to the longest side = 12 cm