Question

find the length of the third side. if necessary, round to the nearest tenth. 24 14

Explanation:

Step1: Identify the triangle type

This is a right - triangle, so we can use the Pythagorean theorem. The Pythagorean theorem states that for a right - triangle with hypotenuse \(c\) and legs \(a\) and \(b\), \(c^{2}=a^{2}+b^{2}\). Here, the hypotenuse \(c = 24\), one leg \(b=14\), and we need to find the other leg \(a\). Rearranging the formula for \(a\), we get \(a=\sqrt{c^{2}-b^{2}}\).

Step2: Substitute the values

Substitute \(c = 24\) and \(b = 14\) into the formula:
\(a=\sqrt{24^{2}-14^{2}}=\sqrt{(24 + 14)(24 - 14)}\) (using the difference of squares formula \(x^{2}-y^{2}=(x + y)(x - y)\))
First, calculate \(24^{2}=576\) and \(14^{2}=196\). Then \(24^{2}-14^{2}=576-196 = 380\). So \(a=\sqrt{380}\).

Step3: Simplify and round

\(\sqrt{380}\approx19.5\) (since \(19.5^{2}=380.25\) which is very close to \(380\))

Answer:

\(19.5\)