The perimeter of a triangle is 450m and its sides are in the ratio of 13:12:5. Find semi perimeter of triangle?
Answers
Hey mate,
Your answer is-
Given :-
Perimeter of the triangle = 450m
➡ It's semi-perimeter = 450/2 = 225m
Ratio between the sides of the triangle is 13 : 12 : 5
Let the sides of the triangle be 13x, 12x and 5x respectively.
We know that,
Perimeter of a triangle = sum of all three sides
➡ 13x + 12x + 5x = 450m
➡ 30x = 450m
➡ x = 450/30
➡ x = 15m
The sides of the triangle are :-
13x = 13 × 15 = 195m
12x = 12 × 15 = 180m
5x = 5 × 15 = 75m
Area of the triangle by heron's formula :-
= √[s(s - a)(s - b)(s - c)] (s = semi-perimeter, a, b and c are the sides respectively)
= √[225(225 - 195)(225 - 180)(225 - 75)
= √(225 × 30 × 45 × 150)
= √(3 × 3 × 5 × 5 × 2 × 3 × 5 × 3 × 3 × 5 × 2 × 3 × 5 × 5)
= √(3 × 3 × 3 × 3 × 3 × 3 × 5 × 5 × 5 × 5 × 5 × 5 × 2 × 2)
= 3 × 3 × 3 × 5 × 5 × 5 × 2
= 6750m²
Now, we've to find the shortest altitude of the triangle.
We also know that,
Area of a triangle = 1/2 × base × height
When base = 195m, 1/2 × 195 × h = 6750m²
➡ h = 6750/1 × 2/195 = 69m (approx)
When base = 180m, 1/2 × 180 × h = 6750m²
➡ h = 6750/9 = 75m
When base = 75m, 1/2 × 75 × h = 6750m²
➡ 6750/37.5 = 180m
Hence, the smallest altitude is 69m.
I hope it will help
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Answer:
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