Q52. - Algebra 1

Question

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

Step-by-Step Solution

Answer

The ladder is $5\sqrt{13}$ feet long

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.