### Video Transcript

Find ๐ฅ in the right triangle
shown.

Weโre 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. Weโve been given the lengths of two
of the triangleโs sides. ๐ผ๐ฝ is 17 units and ๐ผ๐พ is 15
units. ๐ฅ is the length of the third side
in the triangle. And itโs this that weโre 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 and ๐ and the length of the hypotenuse as ๐, this can be
written as ๐ squared plus ๐ squared equals ๐ squared.

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

So weโve found the value of ๐ฅ;
itโs 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.