A non-degenerate right-angled triangle has vertices at \((0, 0), (12, 0)\) and \((0, a)\) with \(a>0 \). Its perimeter is numerically equal to its area. What is \(a\)?
+130114 Area = 12*a / 2 = 6a
Perimeter = a + 12 + sqrt ( a^2 + 12^2)
So
6a = a + 12 + sqrt (a^2 + 12^2)
5a - 12 = sqrt ( a^2 + 144) square both sides
25a^2 - 120a + 144 = a^2 + 144 rearrange as
24a^2 - 120a = 0 factor
24a ( a - 5) = 0
The second factor set = 0 and solved for a gives that a = 5
