## Trigonometry (11th Edition) Clone

A triangle is a perfect triangle if it fulfills two requirements. First, a perfect triangle is a triangle whose sides have whole number lengths. Since the triangle in question has sides 9,10 and 17; it fulfills the first requirement. Second, the area of a perfect triangle is numerically equal to its perimeter. Therefore, we find the perimeter and area of the triangle in question and compare the two: Perimeter= Sum of all three sides Perimeter=$9+10+17$ Perimeter=$36$ To find the area, we need to apply the Heron's formula since the length of all three sides is given: To use the Heron's formula, we first need to find the semi-perimeter $s$. We substitute the values of the sides of the triangle in the formula below to find the semi-perimeter $s$: $s=\frac{1}{2}(a+b+c)$ $s=\frac{1}{2}(9+10+17)$ $s=\frac{1}{2}(36)$ $s=18$ Now, we use the Heron's formula to find the area of the triangle: $A=\sqrt{s(s-a)(s-b)(s-c)}$ $A=\sqrt{18(18-9)(18-10)(18-17)}$ $A=\sqrt{18(9)(8)(1)}$ $A=\sqrt{1296}$ $A=36$ Since the area of the triangle is numerically equal to its perimeter, the triangle also fulfills the second requirement. Therefore, it is a perfect triangle.