Question Video: Solving for the Hypotenuse of a Right-Angled Triangle with Noninteger Solutions Mathematics

A rectangle has side lengths 15 cm and 19 cm. Find the length of its diagonal.

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Video Transcript

A rectangle has side lengths 15 centimeters and 19 centimeters. Find the length of its diagonal.

First, we can sketch a rectangle, and we’ll label the side lengths. One property of rectangles is that all of their interior angles measure 90 degrees. They’re all right angles. The diagonal of a rectangle goes from one vertex across the figure to another vertex. And when we draw this diagonal, we create two congruent right triangles out of our rectangle.

Using what we know about right triangles, we can solve for the length of the diagonal. We can use the Pythagorean theorem, which tells us that the sum of the squares of the two smaller sides will be equal to the larger side squared, which we usually just say π‘Ž squared plus 𝑏 squared equals 𝑐 squared, where 𝑐 is the hypotenuse, the longest side in our triangle. And that is always the side opposite the right angle.

Inside this rectangle, the hypotenuse is the diagonal. For us, that means 15 squared plus 19 squared will equal the diagonal squared. So we square both of those values. 225 plus 361 equals 𝑐 squared, which means 586 is equal to 𝑐 squared. To find out what 𝑐 is equal to, we take the square root of both sides. We’re only interested in the positive square root since we’re dealing with distance. And that means 𝑐 is equal to the square root of 586. Our units here for measuring this distance were centimeters. So we say that the diagonal of this rectangle is the square root of 586 centimeters.

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