Question

each triangle is a $30^\circ$-$60^\circ$-$90^\circ$ triangle, and the hypotenuse of one triangle is the longer leg of an adjacent triangle. the hypotenuse of the largest triangle is $8$ centimeters. what is the number of centimeters in the length of the longer leg of the smallest triangle? express your answer as a common fraction.

Answer 1

Using the properties of $30^\circ$-$60^\circ$-$90^\circ$ triangles to solve the question, the hypotenuse of the largest triangle is twice the length of the longer leg of the smallest triangle. Solving the equation from this ratio gives the longer leg of the smallest triangle as $4$ cm.

Step-by-step explanation:

The student's question involves a series of $30^\circ$-$60^\circ$-$90^\circ$ right triangles, where the hypotenuse of the largest triangle is given as $8$ centimeters, and the hypotenuese of each triangle is the longer leg of the adjacent smaller triangle. In such triangles, the length of the hypotenuse is twice the length of the shorter leg (which is opposite the $30^\circ$ angle), and the longer leg (opposite the $60^\circ$ angle) is \(\sqrt{3}\) times the shorter leg.

Let's denote the length of the longer leg of the smallest triangle as \(x\). The longer leg of the medium triangle is then \(2x\), which then serves as the hypotenuese of the largest triangle. Since the hypotenuse of the largest triangle is given as $8$ cm, we use the ratio properties of $30^\circ$-$60^\circ$-$90^\circ$ triangles to set up the following equation: \(2x = 8\ cm\), which means \(x = 4\ cm\). Therefore, the longer leg of the smallest triangle is \(4\ cm\).