A triangle has a perimeter 13 m. The two shorter sides have integer lengths equal to (x) and (x+1). What could be lengths of each of the 3 sides of the triangle?

1 Answer
Dean R.
May 8, 2018

#(3,4,6), (4,5,4), (5,6,2)#

Explanation:

We know all three triangles have integer sides, since the third sides is the difference of the perimeter and the other two sides, all integers.

So the sides are #x#, #x+1#, and #13-x-(x+1)=12-2x#

The sides have to satisfy the triangle equality, which is the sum of any two side lengths is larger than the third. Let's write the three inequalities:

#x + x + 1 > 12 - 2x#

#4x > 11#

#x > 2 3/4 #

#x + 12-2x > x + 1 #

#2x < 11#

# x < 5 1/2 #

# x + 1 + 12-2x > x#

# 2x < 13 quad # subsumed by the last one

So we have

# 2 3/4 < x < 5 1/2 #

so #x=3,4,5# giving sides #(x,x+1,12-2x)=#

#(3,4,6)#

#(4,5,4)#

#(5,6,2)#

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