Question 534557: The measures of the sides if an isosceles triangle are represented by x + 5, 3x + 13, and 4x + 11. What are the measures of each side of the triangle? Two answers are possible.
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! We know that two of the sides of an isosceles triangle are congruent, meaning that they have the same measure. two of those expressions will have the same value for some x. The problem says two such solutions are possible. I see three approaches two solve the problem. There are probably other ways too. Your choice depends on your knowledge base, your preferences, what you are currently studying, and your teacher's expectations.
TABULATION WITH FINGERS CROSSED APPROACH
We can make a table with 4 columns, one for each of the expressions x, x+5, 3x+13, and 4x+11. We put a zero somewhere in the middle of the x column. We fill 5, 13, and 11 on the same line in the other columns, representing the lengths of the sides of the triangle for the value x=0. As we increase x by 1, we see that the other columns increase by 1, 3, and 4, so they are easy to fill.
At some point we have:
x....x+5...3x+13...4x+11
_...___....____....____
_...___....____....____
_...___....____....____
_...___....____....____
0.......5.......13........11
1........6.......16........15
2.......7.......19........19
3.......8.......22.......23
We realize we found one triangle with side length 7, 19, an 19.
We get to work on filling the top of the table. Keep in mind that negative values in the x column are OK, but negative values in the other column are not. Keep your fingers crossed. If the other answer has side lengths that are whole numbers, we'll find it.
LINEAR EQUATIONS/GRAPHING APPROACH
   are 3 linear equations representing 3 (straight) lines that can be graphed. Each line/linear equation represents the length of one side as a function of x. Each pair of lines will intersect once, giving us a value of y for the equal measures of two sides (not valid if y<0). The x at the intersection point will be used to find the y for the other line (the length of the third side (not valid if y<0).
JUST THE ALGEBRA APPROACH
We write 3 equations to find x (one for each possible unequal side). We will get one solution for each equation.
Then we calculate the measures of the sides for each of the 3 x values found. We are told there are only 2 possible solutions. One of those x values must be an impossible solution because it gives you some negative number for the measure of at least one side.
 -->  --> 
 -->  --> 
 -->  --> 
The first solution gives us one negative length side:
-->  , and is not a solution of the geometry problem.
The other two solutions work:
--> and 
--> and  , because we get positive numbers for the lengths of all 3 sides, and the unequal side is shorter than twice the common length.
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