Video Transcript
Find 𝑥 in the right triangle
shown.
Looking at the information we’ve
been given, we note, first of all, that this triangle is a right triangle. It includes a right angle. And we’ve been given the lengths of
two of its sides. They are eight units and 15
units. 𝑥 represents the length of the
third side of this right triangle. And from its position, directly
opposite the right angle, we note that 𝑥 is the hypotenuse of this triangle. As we’ve been given the lengths of
two sides in a right triangle and we wish to calculate the third, this is exactly
the setup we need in order to apply the Pythagorean theorem.
This tells us that, in a right
triangle, the sum of the squares of the two shorter sides is equal to the square of
the hypotenuse. So we’ll begin by writing down what
the Pythagorean theorem tells us about this triangle in particular. The two shorter sides are eight
units and 15 units. So the sum of the squares of the
two shorter sides is eight squared plus 15 squared. This is then equal to the square of
the hypotenuse. And as the hypotenuse of our
triangle is 𝑥, we now have the equation eight squared plus 15 squared is equal to
𝑥 squared.
So by considering what the
Pythagorean theorem tells us about this triangle in particular, we have an equation
we can solve in order to determine the value of 𝑥. You may prefer to swap the two
sides of the equation around so that 𝑥 is on the left-hand side, although this
isn’t entirely necessary. Now that we formed our equation,
we’re going to solve it by first evaluating eight squared and 15 squared. This gives 𝑥 squared equals 64
plus 225, which simplifies to 𝑥 squared equals 289.
The next step in solving this
equation is to take the square root of each side because the square root of 𝑥
squared will give 𝑥. Now usually, when we solve an
equation by square rooting, we must remember to take plus or minus the square
root. But here 𝑥 has a physical meaning;
it’s the length of a side in a triangle. So it must take a positive
value. We therefore write 𝑥 equals just
the positive square root of 289. 289 is in fact a square number, and
its square root is 17. So we found the value of 𝑥. 𝑥 is equal to 17.
Now, we should always perform a
quick sense check of our answer by comparing the value we found with the other two
sides in the triangle. Remember, 𝑥 represents the
hypotenuse, which is the longest side in this right triangle. So our value for 𝑥 needs to be
bigger than the lengths of the two other sides. Our value is 17 and the two other
sides are 15 and eight. So our answer does make sense.
Now, in fact, this triangle is an
example of a special type of right triangle, called a Pythagorean triple. This is a right triangle in which
all three of the side lengths are integers. The most well-known Pythagorean
triple is the three-four-five triangle as three squared plus four squared is equal
to five squared. You may well encounter Pythagorean
triples when working without a calculator. So it’s a good idea to be familiar
with some of the most common ones. By applying the Pythagorean theorem
then, we’ve found the value of 𝑥 in the right triangle shown is 17.
