Question

compute the perimeter and area of the triangle shown. include units in your computations and answer.
how to enter an answer with units
p = 131 miles

Explanation:

Step1: Find the missing side of the triangle

First, we need to find the length of the left - hand segment of the base. Let's call the left - hand segment of the base \(x\). We can use the Pythagorean theorem in the left - hand right triangle with hypotenuse \(41\) mi and height \(40\) mi. According to the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse and \(a\) and \(b\) are the legs of the right triangle. So, \(x^{2}+40^{2}=41^{2}\). Then \(x^{2}=41^{2}-40^{2}=(41 + 40)(41 - 40)=81\times1 = 81\), so \(x = 9\) mi.

Step2: Calculate the perimeter of the triangle

The sides of the triangle are \(41\) mi, \(50\) mi, and \((9 + 39)\) mi. The perimeter \(P\) of a triangle is the sum of its three sides. So \(P=41+50+(9 + 39)\). First, calculate \(9+39 = 48\) mi. Then \(P=41 + 50+48=139\) mi.

Step3: Calculate the area of the triangle

The area \(A\) of a triangle is given by the formula \(A=\frac{1}{2}\times base\times height\). The base of the triangle is \((9 + 39)=48\) mi and the height is \(40\) mi. So \(A=\frac{1}{2}\times48\times40\). First, \(\frac{1}{2}\times48 = 24\), then \(24\times40 = 960\) square miles.

Answer:

Perimeter \(P = 139\) miles, Area \(A=960\) square miles