SAT Right Triangles & Trigonometry Question 36 | Free SAT Prep | Aniko

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Question 36·Hard·Right Triangles and Trigonometry

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A search-and-rescue team travels from a base camp to a site located due east and due north of the camp, forming a right triangle with the east and north directions as perpendicular legs. The team’s direct route from the camp to the site is 13 kilometers.

The site is 7 kilometers farther east than it is north.

Let $x$ be the angle between the direct route and the east direction. Which choice is the value of $\cos x$ ?

Translate the wording into a right triangle and define variables for the legs when a relationship like “7 more than” is given. Use the Pythagorean theorem to solve for the leg lengths, then compute the requested trig ratio by carefully identifying which side is adjacent to the named angle and which side is the hypotenuse.

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Draw the right triangle from the description

Use east and north as perpendicular legs, and label the direct route as the hypotenuse with length 13.

Use a variable for the northward distance

If the northward distance is $n$ , what expression represents the eastward distance if it is 7 greater?

Match cosine to the correct sides

Since $x$ is measured from the east direction, the side adjacent to $x$ is the eastward leg, and the hypotenuse is the 13-km direct route.

Desmos Guide

Solve for the northward distance

In Desmos, enter n^2 + (n+7)^2 = 13^2 and use the intersection/solution to find the positive value of $n$ .

Compute the eastward distance

Enter n+7 using the positive solution for $n$ .

Form the cosine ratio

Enter (n+7)/13 and simplify the fraction shown; this is $\cos x$ .

Step-by-step Explanation

Represent the legs with a variable

Let the northward distance be $n$ kilometers. Then the eastward distance is $n+7$ kilometers.

The direct route is the hypotenuse of the right triangle, so its length is 13 kilometers.

Use the Pythagorean theorem to solve for $n$

Because the legs are perpendicular,

n^2+(n+7)^2=13^2.

Expand and simplify:

2n^2+14n+49=169

2n^2+14n-120=0

Divide by 2:

n^2+7n-60=0.

Factor:

(n+12)(n-5)=0.

So $n=5$ (reject $n=-12$ since a distance can’t be negative).

Find the eastward leg

Eastward distance $=n+7=5+7=12$ kilometers.

Compute $\cos x$

Angle $x$ is measured from the east direction to the hypotenuse, so the adjacent side is the eastward leg.

\cos x=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{12}{13}.

Therefore, the correct answer is $\frac{12}{13}$ .

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Desmos Guide

Step-by-step Explanation

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Step-by-step Explanation