A rope is stretched from the top of a vertical pole to a point on level ground, forming a right triangle as shown. The pole is meters tall, and the rope is meters long. If is the angle between the rope and the ground at the point where the rope touches the ground, what is the value of ?

Solution available with step-by-step explanation

SAT Right Triangles & Trigonometry Question 20 | Free SAT Prep | Aniko

00:00

Question 20·Medium·Right Triangles and Trigonometry

0 / 50

A rope is stretched from the top of a vertical pole to a point on level ground, forming a right triangle as shown. The pole is $5$ meters tall, and the rope is $13$ meters long. If $\theta$ is the angle between the rope and the ground at the point where the rope touches the ground, what is the value of $\tan(\theta)$ ?

For right-triangle trig problems, first make sure you know which side is opposite and which is adjacent to the given angle. If a leg is missing, use the Pythagorean theorem to find it, then compute $\tan(\theta)=\dfrac{\text{opposite}}{\text{adjacent}}$ (tangent uses only the legs, not the hypotenuse).

Hints

Click me!

Identify the right triangle

The pole is vertical and the ground is horizontal, so the angle at the base of the pole is $90^\circ$ .

Use the Pythagorean theorem

You know the hypotenuse ( $13$ ) and one leg ( $5$ ). Use $a^2+b^2=c^2$ to find the other leg.

Set up tangent correctly

For angle $\theta$ at the ground point, $\tan(\theta)=\dfrac{\text{opposite}}{\text{adjacent}}=\dfrac{\text{pole height}}{\text{horizontal distance}}$ .

Desmos Guide

Compute the missing leg

Enter $x=\sqrt{13^2-5^2}$ to find the horizontal distance.

Form the tangent ratio

Enter $5/x$ to compute $\tan(\theta)$ using opposite over adjacent.

Match to a choice

Compare the value you get to the answer choices to select the match.

Step-by-step Explanation

Find the horizontal distance

The triangle is right, with hypotenuse $13$ and vertical leg $5$ . Let the horizontal leg be $x$ .

By the Pythagorean theorem,

x^2+5^2=13^2

x^2=169-25=144 \implies x=12.

Use tangent as opposite over adjacent

Angle $\theta$ is at the point on the ground, so the opposite side is the pole height ( $5$ ) and the adjacent side is the horizontal distance ( $12$ ).

\tan(\theta)=\frac{5}{12}.

Therefore, the correct answer is $\dfrac{5}{12}$ .

Strategy

Hints

Desmos Guide

Step-by-step Explanation

Study Hints

Strategy

Hints

Desmos Guide

Step-by-step Explanation