A rectangular garden has a straight walking path that connects two opposite corners. The l… — ACT

A rectangular garden has a straight walking path that connects two opposite corners. The length of the path is $170$ meters, and the width of the garden is $80$ meters. What is the length, in meters, of the garden?

A
90
B
120
150Answer
D
188
E
250

Answer

The length of the garden is

150

meters.

The diagonal path, width, and length of the rectangular garden form a right triangle where the path is the hypotenuse. According to the Pythagorean theorem, the square of the length plus the square of the width equals the square of the path:

\text{length}^2 + 80^2 = 170^2

. This simplifies to

\text{length}^2 + 6,400 = 28,900

. Subtracting

6,400

from both sides gives

\text{length}^2 = 22,500

. Taking the square root of

22,500

yields

150

meters.

Step-by-Step Solution

Identify the right triangle formed by the length, width, and diagonal path of the garden.

The width (

80

meters) and the unknown length are the legs, while the diagonal path (

170

meters) is the hypotenuse.

The diagonal of a rectangle forms two congruent right triangles with the rectangle's sides.

Set up the Pythagorean equation to solve for the unknown leg.

\text{length}^2 + 80^2 = 170^2

The Pythagorean theorem states that

a^2 + b^2 = c^2

, where

c

is the hypotenuse.

Calculate the squares of the known lengths.

80^2 = 6,400

and

170^2 = 28,900

Evaluate the exponents to simplify the equation.

Isolate the squared unknown variable.

\text{length}^2 = 28,900 - 6,400 = 22,500

Subtract

6,400

from both sides of the equation.

Take the square root of both sides to find the length.

\text{length} = \sqrt{22,500} = 150

The square root operation reverses the squaring of the variable.

Key Concept

Applying the Pythagorean theorem to find the length of an unknown leg in a right triangle.

Estimated Time:1m 0s

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