Question

in manchester, the library is due south of the courthouse and due west of the community swimming pool. if the distance between the library and the courthouse is 9 kilometers and the distance between the courthouse and the city pool is 10 kilometers, how far is the library from the community pool? if necessary, round to the nearest tenth.

Explanation:

Step1: Identify the right - triangle

The library, courthouse, and community pool form a right - triangle where the distance between the courthouse and the community pool is the hypotenuse ($c = 10$ km) and the distance between the library and the courthouse is one of the legs ($a = 9$ km). We want to find the other leg ($b$).

Step2: Apply the Pythagorean theorem

The Pythagorean theorem is $c^{2}=a^{2}+b^{2}$. Rearranging it to solve for $b$, we get $b=\sqrt{c^{2}-a^{2}}$.
Substitute $a = 9$ and $c = 10$ into the formula: $b=\sqrt{10^{2}-9^{2}}=\sqrt{100 - 81}=\sqrt{19}$.

Step3: Calculate the value

$\sqrt{19}\approx4.4$ (rounded to the nearest tenth).

Answer:

$4.4$