SAT Right Triangles & Trigonometry Question 14 | Free SAT Prep | Aniko

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Question 14·Medium·Right Triangles and Trigonometry

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A guy-wire of length $65$ feet is attached to the top of a vertical mast that is $33$ feet tall and is anchored to level ground at a stake. In the figure shown, what is the value of $\tan A$ , where $A$ is the angle between the guy-wire and the ground at the stake?

For right-triangle trig in a real-world setup, first label which side is opposite and adjacent to the requested angle. If a leg is missing, use the Pythagorean theorem to find it, then compute $\tan$ as $\frac{\text{opposite}}{\text{adjacent}}$ (avoid using the hypotenuse in tangent).

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Identify the right triangle

The mast is vertical and the ground is horizontal, so they form a right angle at the base of the mast.

Find the missing leg

Use the Pythagorean theorem with hypotenuse $65$ and leg $33$ to find the ground distance to the stake.

Set up tangent

At the stake, $\tan A=\dfrac{\text{opposite}}{\text{adjacent}}$ (use the two legs, not the hypotenuse).

Desmos Guide

Compute the ground distance

In Desmos, enter x = sqrt(65^2-33^2) to compute the horizontal distance from the mast to the stake.

Compute the tangent value

Enter 33/x to compute $\tan A$ (opposite over adjacent).

Match to a choice

Match the value of 33/x to the answer choices. The matching choice is $\frac{33}{56}$ .

Step-by-step Explanation

Find the horizontal distance

The mast is perpendicular to the ground, so the triangle is right with hypotenuse $65$ (the wire) and one leg $33$ (the mast). Let the ground distance from the mast to the stake be $x$ .

x^2+33^2=65^2

x^2=65^2-33^2=4225-1089=3136 \Rightarrow x=56.

Use tangent at the stake

Angle $A$ is at the stake. The side opposite $A$ is the mast ( $33$ ), and the side adjacent to $A$ is the ground distance ( $56$ ).

\tan A=\frac{33}{56}.

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation