Question

1. find the perimeter and area of this triangle.

9 mi
24 mi

Explanation:

Step1: Identify the triangle type

The triangle is a right - triangle with legs \(a = 9\) mi and \(b=24\) mi. First, we need to find the hypotenuse \(c\) using the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\).
Substitute \(a = 9\) and \(b = 24\) into the formula: \(c=\sqrt{9^{2}+24^{2}}=\sqrt{81 + 576}=\sqrt{657}=\sqrt{9\times73}=3\sqrt{73}\approx3\times8.544 = 25.632\) mi.

Step2: Calculate the perimeter

The perimeter \(P\) of a triangle is the sum of its three sides. So \(P=a + b + c\).
Substitute \(a = 9\), \(b = 24\) and \(c\approx25.632\): \(P=9 + 24+25.632=58.632\) mi.

Step3: Calculate the area

The area \(A\) of a right - triangle is given by the formula \(A=\frac{1}{2}\times\text{base}\times\text{height}\). Here, the base \(b = 24\) mi and the height \(a = 9\) mi.
So \(A=\frac{1}{2}\times9\times24=9\times12 = 108\) square miles.

Answer:

Perimeter: \(\approx58.63\) mi, Area: \(108\) square miles