The two diagonals of a rectangle are blank. (A) parallel, (B) perpendicular, (C) equal in length, (D) perpendicular and equal in length.
First, let’s think about what we know about rectangles. Rectangles are quadrilaterals, that is, that they are four-sided figures, where all of the interior angles measure 90 degrees. There are four right angles. In a rectangle, opposite sides are parallel. In addition to that, opposite sides are equal in length. If we draw this diagonal, we see that we have created a right triangle. And if we say this side is equal to 𝑎 in length and this side is equal to 𝑏 in length, we know that the two opposite sides will be equal to 𝑎 and 𝑏 in length, respectively.
Now, because this is a right triangle, the diagonal is the hypotenuse of the right triangle. And by the Pythagorean theorem, 𝑎 squared plus 𝑏 squared will equal 𝑐 squared. So we’ll just say the diagonal has a length of 𝑐. But what about the other diagonal? In this case, we’ve created a different right triangle. However, the two shorter sides are still lengths 𝑎 and 𝑏. And since both of these triangles are right triangles and they both have the same smaller sides, the hypotenuse of each of these triangles will be equal in length.
We’re showing that the two diagonals of a rectangle are equal in length to each other. We know that they’re not parallel as they do intersect. However, these diagonals do not intersect at a right angle, which means they’re not perpendicular, and only option (C) can be true. But before we leave this question, there is one notable exception. And that notable exception is the square. A square fits the definition of a rectangle. It’s a special kind of rectangle. And inside a square, the diagonals are equal in length and also perpendicular. For most rectangles, it would be true that the two diagonals are equal in length but not perpendicular. However, for squares, the two diagonals are both perpendicular and equal in length.