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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. A. 7,258.74 ft² B. 1,382.75 ft² C. 11,516.59 ft² D. 9,093.12 ft²

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Answer:

Let's denote the sides of the triangle as \(11x\), \(16x\), and \(24x\) (where \(x\) is a scaling factor).

Given that the perimeter of the triangle is \(510\) feet, we can write the equation:

\[11x + 16x + 24x = 510\]

Solve for \(x\):

\[51x = 510\]

\[x = 10\]

Now we know the lengths of the sides are \(110\), \(160\), and \(240\) feet.

To calculate the area of the triangle using Heron's formula:

\[s = \frac{a + b + c}{2} = \frac{110 + 160 + 240}{2} = 255\]

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

\[A = \sqrt{255 \cdot (255 - 110) \cdot (255 - 160) \cdot (255 - 240)}\]

\[A = \sqrt{255 \cdot 145 \cdot 95 \cdot 15}\]

\[A = \sqrt{814875375}\]

\[A \approx 28,566.38\]

So, the area of the triangle is closest to \(\text{A. 7,258.74 ft²}\).

User Marc Schmid
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