Question

30°-60°-90° triangle theorem: verifying side lengths
30°-60°-90° triangle theorem: in a 30°-60°-90° triangle, the ________ is twice as long as the shorter leg, and the ____ leg is √3 times as long as the shorter leg.

• remember, in triangles, the _____ side is opposite the smallest angle and the next shortest side is opposite the next smallest angle while the longest side is opposite the _____ angle.

image of a right triangle labeled a, b, c with right angle at c, angle at a is 30°, angle at b is 60°, side bc is 1, side ac is √3, side ab is 2

Explanation:

1. For the first blank: In a \(30^\circ - 60^\circ - 90^\circ\) triangle, the hypotenuse is twice the shorter leg. Looking at the triangle, the side labeled 2 is the hypotenuse, and the shorter leg (opposite \(30^\circ\)) would be 1 (since \(2 = 2\times1\)).
2. For the second blank: The longer leg (opposite \(60^\circ\)) is \(\sqrt{3}\) times the shorter leg. The leg labeled \(\sqrt{3}\) is opposite \(60^\circ\), so it's the longer leg (relative to the shorter leg of length 1, since \(\sqrt{3}= \sqrt{3}\times1\)).
3. For the third blank: In any triangle, the shortest side is opposite the smallest angle. The smallest angle here is \(30^\circ\), so the shortest side is opposite it.
4. For the fourth blank: The longest side is opposite the largest angle. In a \(30^\circ - 60^\circ - 90^\circ\) triangle, the largest angle is \(90^\circ\), so the longest side (hypotenuse) is opposite the right (or \(90^\circ\)) angle.

Answer:

First blank: hypotenuse; Second blank: longer; Third blank: shortest; Fourth blank: largest (or right/90°)