Question

find p.
right triangle with right angle, one angle 60°, one angle 30°, leg 16√3 ft, other leg p
write your answer in simplest radical form.
blank feet
√ button
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Explanation:

Step1: Identify the triangle type

This is a 30 - 60 - 90 right triangle. In a 30 - 60 - 90 triangle, the ratios of the sides are \(1:\sqrt{3}:2\) (opposite 30°, 60°, 90° respectively). Let's denote the sides: the side opposite 30° is the shortest side, opposite 60° is \( \sqrt{3}\) times the shortest side, and the hypotenuse is twice the shortest side.

Looking at the triangle, the side with length \(16\sqrt{3}\) ft is opposite the 60° angle, and \(p\) is opposite the 30° angle.

Step2: Use the ratio of sides in 30 - 60 - 90 triangle

Let the side opposite 30° (which is \(p\)) be \(x\), the side opposite 60° be \(x\sqrt{3}\), and the hypotenuse be \(2x\).

We know that the side opposite 60° is \(16\sqrt{3}\) ft. So, \(x\sqrt{3}=16\sqrt{3}\)

To find \(x\) (which is \(p\)), we divide both sides of the equation \(x\sqrt{3}=16\sqrt{3}\) by \(\sqrt{3}\):

\(x=\frac{16\sqrt{3}}{\sqrt{3}}\)

Simplifying the right - hand side, \(\sqrt{3}\) in the numerator and denominator cancels out, so \(x = 16\)

Answer:

\(16\)