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

1 answer:

5 0

When the sides are in a ratio of 11:16:24, it suggests they are multiples of a common variable x, based on these proportions.

The shortest side measures 11x feet, the middle side is 16x feet, and the longest measures 24x feet.

This indicates a perimeter of

$11x+16x+24x = 51x$ feet. Given that the actual perimeter is 510 feet, we have:

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

Thus, the sides measure 110, 160, and 240 feet each.

To determine the area of the triangle with these three sides, Heron's formula is applicable. This formula states that, if $s$ represents half of the perimeter for a triangle with sides $a,b,c$ , the area $A$ is given by:

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

In this scenario, $s = 255, a = 110, b = 160, c = 240$ , so the formula adapts to:

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

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