Question

the longer leg of a 30°-60°-90° triangle has a length of 45 meters. what is the length of its hypotenuse?
write your answer in simplest radical form.
□ meters
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Explanation:

Step1: Recall 30-60-90 triangle ratios

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: Relate longer leg to \(x\)

Given the longer leg is \(45\) meters, and longer leg \(= x\sqrt{3}\). So, \(x\sqrt{3}=45\). Solve for \(x\): \(x = \frac{45}{\sqrt{3}}\). Rationalize the denominator: \(x=\frac{45\sqrt{3}}{3}=15\sqrt{3}\). Wait, no, wait. Wait, hypotenuse is \(2x\)? Wait no, wait: shorter leg is \(x\), longer leg is \(x\sqrt{3}\), hypotenuse is \(2x\). Wait, but we can also relate longer leg to hypotenuse directly. Let hypotenuse be \(h\). Then, \(\sin60^\circ=\frac{\text{longer leg}}{h}\). \(\sin60^\circ = \frac{\sqrt{3}}{2}\), so \(\frac{\sqrt{3}}{2}=\frac{45}{h}\). Cross - multiply: \(h\sqrt{3}=90\), so \(h = \frac{90}{\sqrt{3}}\). Rationalize: \(h=\frac{90\sqrt{3}}{3}=30\sqrt{3}\)? Wait, no, wait. Wait, let's do it with the ratio. Longer leg is \(x\sqrt{3}\), hypotenuse is \(2x\). So if \(x\sqrt{3}=45\), then \(x = \frac{45}{\sqrt{3}}\), then hypotenuse \(2x=\frac{90}{\sqrt{3}}=\frac{90\sqrt{3}}{3}=30\sqrt{3}\). Wait, that's correct? Wait, no, wait. Wait, in 30-60-90 triangle, the sides are: shorter leg (opposite 30) = \(x\), longer leg (opposite 60) = \(x\sqrt{3}\), hypotenuse = \(2x\). So if longer leg is \(x\sqrt{3}=45\), then \(x=\frac{45}{\sqrt{3}}\), then hypotenuse is \(2x = \frac{90}{\sqrt{3}}\). Multiply numerator and denominator by \(\sqrt{3}\): \(\frac{90\sqrt{3}}{3}=30\sqrt{3}\). Yes, that's correct.

Wait, another way: Let hypotenuse be \(h\). Then, \(\cos30^\circ=\frac{\text{longer leg}}{h}\)? No, \(\cos30^\circ=\frac{\text{adjacent}}{\text{hypotenuse}}\), adjacent to 30 is longer leg? Wait, no. Angle 30: adjacent side is longer leg, opposite is shorter leg. So \(\cos30^\circ=\frac{\text{longer leg}}{h}\), \(\cos30^\circ=\frac{\sqrt{3}}{2}\), so \(\frac{\sqrt{3}}{2}=\frac{45}{h}\), so \(h=\frac{90}{\sqrt{3}} = 30\sqrt{3}\). Yes.

Answer:

\(30\sqrt{3}\)