Seeking The Right Answer
A non-degenerate right-angled triangle has vertices at and with . Its perimeter is numerically equal to its area. What is ?
The answer is 5.
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1 solution
Moderator note:
Interesting question.
We can take any given triangle and scale it, to make the perimeter numerically equal to the area.
The legs of the triangle have length 12 and a. Since the area is equal to the perimeter:
1 / 2 × 1 2 × a = 1 2 + a + 1 2 2 + a 2
6 a = 1 2 + a + 1 4 4 + a 2
5 a − 1 2 = 1 4 4 + a 2
2 5 a 2 − 1 2 0 a + 1 4 4 = 1 4 4 + a 2
2 4 a 2 − 1 2 0 a = 0
( 2 4 a ) ( a − 5 ) = 0
24a = 0 or a-5 = 0
a = 0 or a = 5
Since the triangle is not degenerate, we eliminate a = 0, leaving a = 5.