Question

1. choose the correct answer. apply the 30°-60°-90° triangle theorem to find the length of the longer leg of a triangle if the length of the hypotenuse is 20 cm. 17 cm \\(\sqrt{5\sqrt{20}}\\) cm \\(\sqrt{20}\\) cm 10 cm

Explanation:

Step1: Recall 30-60-90 Triangle Theorem

In a \(30^\circ - 60^\circ - 90^\circ\) triangle, the sides are in the ratio \(1 : \sqrt{3} : 2\), where the side opposite \(30^\circ\) (shorter leg) is \(x\), the side opposite \(60^\circ\) (longer leg) is \(x\sqrt{3}\), and the hypotenuse is \(2x\).

Step2: Find the shorter leg (\(x\))

Given the hypotenuse is \(20\) cm. From the ratio, hypotenuse \(= 2x\). So, \(2x = 20\). Solving for \(x\), we divide both sides by \(2\): \(x=\frac{20}{2}=10\) cm. This is the shorter leg.

Step3: Find the longer leg

The longer leg (opposite \(60^\circ\)) is \(x\sqrt{3}\). Substituting \(x = 10\), we get longer leg \(= 10\sqrt{3}\) cm? Wait, wait, looking at the options, maybe I misread. Wait, the options: let's check again. Wait, the options are 17 cm, \(\sqrt{5\sqrt{20}}\) cm, \(\sqrt{20}\) cm, 10 cm? Wait, no, maybe the hypotenuse is 20? Wait, no, maybe the hypotenuse is 20? Wait, no, wait, maybe the problem has a typo, or I misread. Wait, no, in a 30-60-90 triangle, hypotenuse is twice the shorter leg. If hypotenuse is 20, shorter leg is 10, longer leg is \(10\sqrt{3}\approx17\) cm. Ah, so the first option is 17 cm (approximate, since \(10\sqrt{3}\approx17.32\approx17\)).

Wait, let's recalculate:

Hypotenuse \(c = 20\) cm.

Shorter leg \(a=\frac{c}{2}=\frac{20}{2}=10\) cm.

Longer leg \(b = a\sqrt{3}=10\sqrt{3}\approx17.32\approx17\) cm. So the correct option is 17 cm.

Answer:

17 cm (the first option, assuming the options are labeled as: first option 17 cm, etc.)