Two different triangles have the same area but different perimeters. Triangle has side lengths of and cm. Triangle has two sides with length of cm. Find the length of the third side of triangle . Round your answer to the nearest tenth of a cm.

Source: mathcontest.olemiss.edu 9/9/2013

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SOLUTION
**Since triangle is isosceles, altitude is the perpendicular bisector of the base.

Area of triangle

Since triangle is isosceles, altitude is the perpendicular bisector of the base.

Area of triangle = area of triangle

Substitute the value of into Eq.

The equation is quadratic in

First solution

Length of third side

cm

Second solution

Length of third side

cm which is the length of the base of triangle .

**Answer**: cm