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Video: Applying the Converse of Pythagoras’s Theorem

Kathryn Kingham

A triangle has sides of lengths 36.4, 27.3, and 45.5. What is its area?


Video Transcript

A triangle has side lengths 36.4, 27.3, and 45.5. What is its area?

We know that to find the area of a triangle, we multiply the height times the base and then divide that by two. A base can be any side of the triangle, but the height must be perpendicular to the base that we choose. That means that for whatever base we choose, the height must form a right angle with that base. But how do we know if any of these lines form a right angle with each other? Great question! We could use the Pythagorean theorem to check and see if these three side lengths form a right triangle. If they do form a right triangle, then sides a and b are perpendicular to each other, and we can use them to find the area. So let’s start there.

Let’s plug in these side lengths and see if this is a right triangle. 36.4 squared plus 27.3 squared equals 45.5 squared. Notice that I put the longest side length in the c position. If this is a right triangle, then the hypotenuse will be across from the right angle and will be the longest side. 36.4 squared equals 1324.96, 27.3 squared equals 745.29, 45.5 squared equals 2070.25. If we add the a squared and the b squared, we get 2070.25.

What this does for us is it confirms that we are working with a right triangle. If we sketch our triangle and label our sides, we recognize that we could use 27.3 as a height and 36.4 as the base for our triangle. We plug in the height and the base to the formula for finding the area of our triangle. 27.3 times 36.4 equals 993.72. When we divide that by two, we get 496.86. Since we’re dealing with area, this would be units squared.

Area equals 496.86 units squared.