Question

A ladder is leaning on a building. The base of the ladder is 10 feetfrom the building, and the ladder reaches up 15 feeton the building. How long is the ladder?

Short answer

The ladder is$5\sqrt{13}$ feet long

Step-by-step solution

Step 1. Given. 

The base of the ladder$=10\mathrm{feet}$

The ladder reaches up to a height of 15 feet

Step 2. Draw the rough sketch for the given problem.

The rough sketch for the given problem is:

$AC$is the building and$CE$ is the ground.

$BC$is the height up to which ladder can reach on the building.

$CD$is the distance of the base of the ladder from the building.

$BD$is the ladder.

Step 3. Determine the length of the ladder.

In the triangle $BCD$,$\angle BCD$ is a right angle. Therefore, the triangle$BCD$ is right-angled.

According to the Pythagorean theorem, it can be noticed that:

${\left(BD\right)}^2={\left(BC\right)}^2+{\left(CD\right)}^2$

Therefore,

$$\begin{gathered} {\left(BD\right)}^2={\left(BC\right)}^2+{\left(CD\right)}^2 \\ {\left(BD\right)}^2={\left(15\right)}^2+{\left(10\right)}^2 \\ {\left(BD\right)}^2=225+100 \\ {\left(BD\right)}^2=325 \\ BD=\sqrt{325} \\ BD=\sqrt{25\times 13} \\ BD=\sqrt{5^2\times 13} \\ BD=5\sqrt{13} \end{gathered}$$

Therefore, the ladder is$5\sqrt{13}$ feet long.