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Question Video: Finding the Side Length of an Equilateral Triangle Using Heron’s Formula Mathematics

The area of an equilateral triangle is 947√3 cm². Find the side length using Heron’s formula giving the answer to the nearest centimeter.

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Video Transcript

The area of an equilateral triangle is 947 times the square root of three centimeters squared. Find the side length using Heron’s formula, giving the answer to the nearest centimeter.

Since our triangle is equilateral, all three sides have the same length, so let’s call this length 𝑥. We’re given the area of the triangle, that’s 947 times root three centimeters squared, and asked to use Heron’s formula to calculate the side length which we’ve labeled 𝑥. This tells us that for a triangle with side lengths 𝑎, 𝑏, and 𝑐, the area 𝐴 is equal to the square root of 𝑠 times 𝑠 minus 𝑎 times 𝑠 minus 𝑏 times 𝑠 minus 𝑐, where 𝑠 is the semiperimeter of the triangle; that’s the sum of the side lengths over two. Now in our case, 𝑎, 𝑏, and 𝑐 are all equal to 𝑥, so the semi perimeter 𝑠 is 𝑥 plus 𝑥 plus 𝑥 over two; that’s three 𝑥 over two.

In our formula for the area 𝐴, we have 𝐴 equal to the square root of 𝑠 times 𝑠 minus 𝑥 times 𝑠 minus 𝑥 times 𝑠 minus 𝑥. That’s the square root of 𝑠 times 𝑠 minus 𝑥 cubed. We already have 𝑠 in terms of 𝑥, and remember it’s 𝑥 we want to find. So substituting 𝑠 equals three over two 𝑥 into our expression for 𝐴, we have 𝐴 equal to the square root of three over two times 𝑥 times three over two 𝑥 minus 𝑥 cubed. Now, three over two 𝑥 minus 𝑥 is equal to 𝑥 over two. And so, our area is the square root of three over two times 𝑥 times 𝑥 over two cubed. Using the power law for indices on our cube inside the square root, we have the square root of three over two 𝑥 multiplied by 𝑥 cubed over two cubed, which simplifies to the square root of three over two to the power four times 𝑥 to the power four.

And now making a little more room and using the fact that by the laws of indices, the positive square root of a number or expression to the fourth power is that number or expression squared, and now since area is always positive, we have the area equal to root three over two squared multiplied by 𝑥 squared. That’s root three over four times 𝑥 squared. Now remember, we’re actually given that the area is equal to 947 root three centimeters squared. And if we divide both sides by root three and multiply both sides by four, we have 𝑥 squared is equal to 947 times four. That’s 𝑥 squared is equal to 3788.

Taking the positive square root on both sides, since length is always positive, we have 𝑥 equal to the square root of 3788, which to four decimal places is 61.5467. We’re asked for the side length to the nearest centimeter, and that’s 62 centimeters. Hence, the side length to the nearest centimeter of the equilateral triangle with area equal to 947 root three centimeters squared is 62.

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