Question

97 km, 65 km, b. what is the perimeter? if necessary, round to the nearest tenth. kilometers

Explanation:

Step1: Identify the triangle type

This is a right - triangle, so we can use the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse, and \(a\) and \(b\) are the legs. Here, \(c = 97\space km\) and \(a=65\space km\), we need to find \(b\).
Rearranging the Pythagorean theorem to solve for \(b\), we get \(b=\sqrt{c^{2}-a^{2}}\)

Step2: Calculate the length of \(b\)

Substitute \(c = 97\) and \(a = 65\) into the formula:
\(b=\sqrt{97^{2}-65^{2}}=\sqrt{(97 + 65)(97 - 65)}\) (using the difference of squares formula \(x^{2}-y^{2}=(x + y)(x - y)\))
\(97+65 = 162\), \(97 - 65=32\)
So \(b=\sqrt{162\times32}=\sqrt{5184}=72\space km\)

Step3: Calculate the perimeter

The perimeter \(P\) of a triangle is the sum of its three sides. So \(P=a + b + c\)
Substitute \(a = 65\), \(b = 72\), \(c=97\) into the formula:
\(P=65 + 72+97=234\space km\)

Answer:

\(234\)