Question Video: Applying the Pythagorean Theorem to Calculate One of the Legs of a Triangle | Nagwa Question Video: Applying the Pythagorean Theorem to Calculate One of the Legs of a Triangle | Nagwa

Question Video: Applying the Pythagorean Theorem to Calculate One of the Legs of a Triangle Mathematics • First Year of Preparatory School

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Find ๐‘ฅ in the right triangle shown.

02:16

Video Transcript

Find ๐‘ฅ in the right triangle shown.

Weโ€™re told that this is a right triangle. And we can also identify this from the figure as one of the interior angles has been marked with a small square. Weโ€™ve been given the lengths of two of the triangleโ€™s sides. ๐ผ๐ฝ is 17 units and ๐ผ๐พ is 15 units. ๐‘ฅ is the length of the third side in the triangle. And itโ€™s this that weโ€™re asked to calculate.

We can recall that whenever we know the lengths of two sides of a right triangle, we can calculate the length of the third side by applying the Pythagorean theorem. This states that in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the two shorter sides. If we label the lengths of the two shorter sides as ๐‘Ž and ๐‘ and the length of the hypotenuse as ๐‘, this can be written as ๐‘Ž squared plus ๐‘ squared equals ๐‘ squared.

In our triangle, ๐‘ฅ is one of the shorter sides. So we can form the equation ๐‘ฅ squared plus 15 squared equals 17 squared. We now want to solve this equation to find the value of ๐‘ฅ. Evaluating the squares gives ๐‘ฅ squared plus 225 equals 289. We can then isolate the ๐‘ฅ squared term by subtracting 225 from each side of the equation, to give ๐‘ฅ squared equals 64. Finally, we take the square root of both sides, giving ๐‘ฅ equals the square root of 64, which is eight. Weโ€™re only interested in the positive solution here as ๐‘ฅ represents a length.

So weโ€™ve found the value of ๐‘ฅ; itโ€™s eight. Interestingly, as all the side lengths in this right triangle are integer values, this is an example of a Pythagorean triple, that is, a right triangle in which all three side lengths are integers. Being able to recognize Pythagorean triples can speed up some problems like these.

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