Question

The perimeter of a triangle is 510 feet and the sides are in the ratio of 11:16:24. Find the area of the triangle

Answer 1

If the sides are in the ratio of 11:16:24, it means that they are all multiples of a same number x, according to these factors.

So, the shorter side is 11x feet long, the middle one is 16x feet long, and the longest side is 24x feet long.

This means that the perimeter is

$11x+16x+24x = 51x$ feet long. But we know that this is 510 feet, so we have

$51x = 510 \implies x = \cfrac{510}{51} = 10$.

So, the three sides are 110, 160 and 240 feet long.

To find the area of a triangle knowing its three sides, you can use Heron's formula, which states that, if $s$ is half the perimeter of the triangle whose sides are $a,b,c$, the area $A$ is given by

$A = \sqrt{s(s-a)(s-b)(s-c)}$

In our case, $s = 255, a = 110, b = 160, c = 240$ so the formula becomes

$A = \sqrt{255(255-110)(255-160)(255-240)} = \sqrt{255(145)(95)(15)} = \sqrt{52689375} \approx 7258.74$

Answer 2

The area of the triangle is calculated from the given dimensions as 7,258.75 ft²

Given:

the perimeter of the triangle, P = 510 ft

ratio of the sides of the triangle, = 11:16:24

To find:

• the area of the triangle

Let the common factor of the sides = y

Then the perimeter is calculated as;

11y + 16y + 24y = 510

51y = 510

y = 510/51

y = 10

The first side of the triangle, a = 11 x 10 = 110 ft

The second side of the triangle, b = 16 x 10 = 160 ft

The third side of the triangle, c = 24 x 10 = 240 ft

The area of the triangle is calculated as by applying Heron's formula;

$Area = \sqrt{s(s-a)(s-b)(s-c)} \\\\s = \frac{a + b+ c}{2} = \frac{110 + 160 + 240}{2} = 255\\\\Area = \sqrt{255(255-110)(255-160)(255-240)}\\\\Area = \sqrt{255(145)(95)(15)} \\\\Area= 7,258.75 \ ft^2$

Thus, the area of the given triangle is 7,258.75 ft²

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