Question Video: Finding the Area of a Square Drawn on a Right Triangle Mathematics • 9th Grade

The length of the hypotenuse of a right triangle is 64 cm and the length of one of the other sides is 59 cm. Find the area of the square drawn on that unknown side.

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Video Transcript

The length of the hypotenuse of a right triangle is 64 centimeters and the length of one of the other sides is 59 centimeters. Find the area of the square drawn on that unknown side.

In this question, we want to find the area of a square drawn from a side of a right triangle. We are told that the hypotenuse of the right triangle has length 64 centimeters and another side has length 59 centimeters. To answer this question, it is a good idea to sketch the given information to help us understand what is being asked.

We start by sketching a right triangle. The hypotenuse is 64 centimeters; that is the longest side of the right triangle. We know that this side is opposite the right angle. We can also label another sideβs length as 59 centimeters. If we say that the length of the unknown side of the triangle is π₯ centimeters, then we can sketch a square drawn from that unknown side. We know that the area of the square is π₯ squared square centimeters.

We can find an expression involving π₯ by recalling that the Pythagorean theorem tells us that the square of the length of the hypotenuse must be equal to the sum of the squares of the lengths of the two shorter sides. Therefore, we have 64 squared is equal to 59 squared plus π₯ squared. Remember, the area of the square is given by π₯ squared square centimeters, so we do not need to solve this equation for π₯. We only need to find the value of π₯ squared. We can then subtract 59 squared from both sides of the equation to find that π₯ squared is equal to 64 squared minus 59 squared.

We could evaluate this expression using a calculator. However, we can simplify by using the formula for factoring a difference of two squares. We recall that this tells us that π squared minus π squared is equal to π minus π times π plus π. Setting π equal to 64 and π equal to 59 gives us that π₯ squared is equal to 64 minus 59 multiplied by 64 plus 59. We can then evaluate the expression in each factor to obtain five times 123 is equal to π₯ squared. We can then calculate that five times 123 is equal to 615.

Finally, we can include the units to conclude that the area of the square drawn on the unknown side of the right triangle is 615 square centimeters.