Question

find the length of the third side. if necessary, round to the nearest tenth.

Explanation:

Step1: Apply Pythagorean theorem

Let the third side be \(x\). According to the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\) (for a right - triangle, where \(c\) is the hypotenuse). Here, \(a = 7\), \(c=14\), and we want to find \(b\). So the formula becomes \(b=\sqrt{c^{2}-a^{2}}\).
Substitute \(a = 7\) and \(c = 14\) into the formula: \(b=\sqrt{14^{2}-7^{2}}\).

Step2: Calculate the values inside the square root

First, calculate \(14^{2}=196\) and \(7^{2} = 49\). Then \(14^{2}-7^{2}=196 - 49=147\). So \(b=\sqrt{147}\).

Step3: Simplify the square root

We know that \(147=49\times3\), so \(\sqrt{147}=\sqrt{49\times3}\). Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (\(a = 49\), \(b = 3\)), we get \(\sqrt{49\times3}=\sqrt{49}\times\sqrt{3}=7\sqrt{3}\approx7\times1.732 = 12.124\). Rounding to the nearest tenth, \(b\approx12.1\).

Answer:

\(12.1\)