Two noncongruent triangles have the same area but different perimeters. Triangle A has sides lengths of 8, 8 and 6 cm. Triangle B has two sides with length of 8 cm. Find the length of the third side of Triangle B. Round your answer to the nearest hundredth of a cm.

Source: mathcontest.olemiss.edu 10/2/2006

**SOLUTION**

In isosceles , altitude is also the perpendicular bisector of base .

In right ,

Area of

In isosceles , altitude is also the perpendicular bisector of base .

Let .

Area of

Since the two triangles have the same area

In right ,

multiply by

(1)

Let . Then, Eq.(1) has the following quadratic form

factoring the equation

There are two solutions

and

If , then which implies that . In this case,

. We reject this solution because the two triangles are noncongruent.

If , then which implies that . In this case,

**Answer**:

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Retired high school math teacher.