In the diagram of parallelogram HIJK below, the diagonals HJ and IK are perpendicular, as shown. If HJ is 10 inches long and IK is 24 inches long, then why must HK be 13 inches long?

It is the hypotenuse of a right triangle with legs 5 and 12.

Step-by-step explanation:

The diagonals of a parallelogram bisect each other. If the angle at which they meet is a right angle, then a right triangle is formed whose legs are half the length of each of the diagonals, and whose hypotenuse is the length of one side of the parallelogram (rhombus).

Here, the triangle's leg lengths are 5 and 12, so the Pythagorean theorem tells us the hypotenuse is ...

√ (5²+12²) = √ (25+144) = √169 = 13

Since the perpendicular diagonals are 10 and 24 inches long, the side length of the rhombus must be 13 inches long.

HK must be 13 inches because the Pythagorean theorem applies and that's what it says.