Question

Two ladders rest against a wall at the same angles. The first ladder is 13 feet long, has its base 5 ft away from the wall and the top touches the wall at a height of 12 feet. The second ladder is 6.5 feet long and has a height of 6ft. How far is the base from the wall?

Answer 1

2.5 feet

Explanation:

Given two ladders resting against a wall at the same angle, and various lengths associated with the first ladder, you want to know how far from the wall is the base of the second ladder.

Similar triangles

The wall and the ground form a right angle, so the fact that the ladders make the same angles with the wall and ground means the geometry can be modeled by similar triangles. Similar triangles have proportional sides.

To find the distance of the second ladder from the wall, we can use the proportion ...

(distance from wall)/(ladder length) = x/(6.5 ft) = (5 ft)/(13 ft)

Solution

Multiplying this proportion by 6.5 ft gives ...

x = (6.5 ft)(5/13) = 2.5 ft

The base of the second ladder is 2.5 feet from the wall.