Answer
The triangle has whole number lengths and its area and perimeter is both 36. Therefore, it is a perfect triangle.
Work Step by Step
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.
