# ALGEBRA

The perimeter of a triangle is 52 centimeters. The longest is 2 centimeters less than the sum of the other 2 sides. Twice the shortest side is 13 centimeters less than the longest side. Find the lengths of each side of the triangle. What are the lengths of the sides from shortest to longest?

1. Reiny

shortest --- x
middle -----y
longest ----z

x+y+z= 52

" Twice the shortest side is 13 centimeters less than the longest side" ---> 2x < z by 13
2x + 13 = z

"The longest is 2 centimeters less than the sum of the other 2 sides" ---> z < x+y by 2
z = x+y - 2
y = z-x+2 = (2x+13) - x + 2
= x + 15

in the 1st:
x+y+z=52
x+(x+15) + (2x+13) = 52
4x = 52-28
x = 6 , then y = 21 and z = 25
The sides are 6, 21 and 25

is the sum equal to 52 ? 6+21+25 = 52 , YES

twice the shortest = 12
is 12 less than 25 by 13 ? , YES

longest = 25
2 less than the sum of the other two = 6+21-2 = 25 , YES
All is good

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