Question

in bluepoint, 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 8 miles and the distance between the courthouse and the city pool is 17 miles, how far is the library from the community pool? miles

Explanation:

Step1: Identify the right - triangle

Since the library is due south of the courthouse and due west of the community swimming pool, the three locations form a right - triangle. Let the distance between the library and the courthouse be \(a = 8\) miles, the distance between the courthouse and the community pool be \(c=17\) miles, and the distance between the library and the community pool be \(b\). By the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), we can rewrite it to find \(b\) as \(b=\sqrt{c^{2}-a^{2}}\).

Step2: Substitute the values

Substitute \(a = 8\) and \(c = 17\) into the formula \(b=\sqrt{c^{2}-a^{2}}\). So \(b=\sqrt{17^{2}-8^{2}}=\sqrt{(17 + 8)(17 - 8)}\) (using the difference of squares \(x^{2}-y^{2}=(x + y)(x - y)\)). Then \(b=\sqrt{25\times9}\).

Step3: Calculate the square - root

\(\sqrt{25\times9}=\sqrt{25}\times\sqrt{9}\) (using the property \(\sqrt{xy}=\sqrt{x}\cdot\sqrt{y},x\geq0,y\geq0\)). Since \(\sqrt{25}=5\) and \(\sqrt{9}=3\), then \(b = 15\).

Answer:

15