Question

Perimeter of a triangle is 450 m and its sides are in the ratio 13 : 12:5. Find the area of the
triangle and smallest altitude.​

Answer 1

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.

Answer 2

Answer :-

Area of triangle = 6750 m²

& Smallest altitude = 69 m .

Solution :-

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 = 450

➞ 30x = 450

➞ x = 450/30

➞ x = 15 m

Hence, the sides of the triangle are :-

13x = 13 × 15 = 195 m

12x = 12 × 15 = 180 m

5x = 5 × 15 = 75 m

By Heron's Formula formula :-

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

= √[225(225 - 195)(225 - 180)(225 - 75)]

= √(225 × 30 × 45 × 150)

= √(225 × 2025 × 100)

= 15 × 45 × 10

= 6750 m²

∴ Area of triangle = 6750 m²

Now, we've to find the shortest altitude of the triangle.

We have,

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

when base = 195 m,

then, 6750 = 1/2 × 195 × h

➞ h = 6750/97.5 = 69.23 ≈ 69 m

when base = 180 m,

then, 6750 = 1/2 × 180 × h

➞ h = 6750/9 = 75 m

when base = 75 m,

then, 6750 = 1/2 × 75 × h

➞ h = 6750/37.5 = 180 m

Hence, the smallest altitude is 69 m.

______________________[Answer]