Question

a ladder is leaning against a wall. the base of the ladder is 4 feet away from the base of the wall, and the ladder reaches a height of 12 feet on the wall. how long is the ladder? if necessary, express your answer as a simplified square root.
○ $4\sqrt{10}$ feet
○ $16\sqrt{10}$ feet
○ 160 feet
○ 16 feet

Explanation:

Step1: Identify the right triangle

The ladder, wall, and ground form a right triangle. The height on the wall (\( a = 12 \) ft) and the distance from the wall (\( b = 4 \) ft) are the legs, and the ladder (\( c \)) is the hypotenuse.

Step2: Apply the Pythagorean theorem

The Pythagorean theorem is \( a^2 + b^2 = c^2 \). Substitute \( a = 12 \) and \( b = 4 \):
\( 12^2 + 4^2 = c^2 \)
\( 144 + 16 = c^2 \)
\( 160 = c^2 \)

Step3: Simplify the square root

Take the square root of both sides: \( c = \sqrt{160} \). Simplify \( \sqrt{160} \) by factoring out the perfect square \( 16 \):
\( \sqrt{160} = \sqrt{16 \times 10} = \sqrt{16} \times \sqrt{10} = 4\sqrt{10} \).

Answer:

A. \( 4\sqrt{10} \) feet