A hiker starts at a trailhead, point A, and walks 9 miles due east, then 12 miles due nort… — ACT

A hiker starts at a trailhead, point $A$ , and walks $9$ miles due east, then $12$ miles due north to reach a campsite, point $C$ . A lookout tower, point $T$ , is located due west of the campsite $C$ . The straight-line distance from the starting point $A$ to the tower $T$ is $13$ miles. If the tower $T$ is located east of the north-south line passing through point $A$ , what is the distance, in miles, between the campsite $C$ and the tower $T$ ?

Answer: 4 miles

Answer

The distance between the campsite and the tower is 4 miles.

The correct answer is 4 miles. By modeling the hiker's path, the vertical height of both the campsite and the tower is 12 miles north of the starting point's east-west line. A right triangle is formed by the trailhead, the projection of the tower onto the east-west axis, and the tower itself. The hypotenuse is 13 miles and the vertical leg is 12 miles. By the Pythagorean theorem, the horizontal leg is 5 miles. Since the tower is east of the trailhead's north-south line, it is 5 miles east of the trailhead. The campsite is 9 miles east of the trailhead. The horizontal distance between the campsite and the tower is the difference: 9 - 5 = 4 miles.

Step-by-Step Solution

Determine the vertical height of the campsite and the tower.

The campsite

C

12

miles north of the trailhead

A

's east-west line. Since the tower

T

is located due west of

C

, it lies on the same horizontal line. Therefore, the vertical distance from the east-west line to the tower

T

is also

12

miles.

Points on the same due east-west line share the same vertical offset (latitude) from the reference axis.

Use the Pythagorean theorem to find the horizontal distance from the trailhead to the tower.

Let

D

be the point on the east-west line directly below the tower

T

. A right triangle

\triangle ADT

is formed where the vertical leg

DT = 12

miles, the hypotenuse

AT = 13

miles, and the horizontal leg is

AD

. Using the Pythagorean theorem:

AD^2 + DT^2 = AT^2 \implies AD^2 + 12^2 = 13^2 \implies AD^2 + 144 = 169 \implies AD^2 = 25 \implies AD = 5

miles.

Calculating the horizontal offset of the tower from the trailhead's north-south line.

Calculate the horizontal distance between the campsite and the tower.

The campsite

C

9

miles east of the trailhead

A

's north-south line, and the tower

T

5

miles east of it. The horizontal distance between them is the difference:

9 - 5 = 4

miles.

Since both points have the same vertical coordinate, the direct distance is simply the difference in their horizontal coordinates.

Key Concept

Using the Pythagorean theorem to solve multi-step geometry problems on a coordinate-like plane.

Estimated Time:1m 30s

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