How to Find the Third Side of an Isosceles Triangle Given Two Equal Sides and the Area

Understanding the geometry of an isosceles triangle and using the triangle area formula can help you to find the length of the third side. This guide will walk you through the steps to calculate the length of the base when you are given the lengths of the two equal sides and the area of the triangle.

Understanding the Triangle

Two Equal Sides: Let the equal sides of the isosceles triangle be denoted by a. The Base: The third side, also known as the base, will be denoted by b. The Height: The height, which can be found using the area formula, will be denoted by h.

The Area Formula

The area of a triangle can be expressed as:

A (frac{1}{2}) × base × height

For our isosceles triangle, this becomes:

A (frac{1}{2}) × b × h

Finding the Height

The height h can be found using the Pythagorean theorem. The height divides the base b into two equal segments of length (frac{b}{2}). Thus we can express the height as:

h (sqrt{a^2 - left(frac{b}{2}right)^2})

Substituting the Height

Substitute h back into the area formula:

A (frac{1}{2}) × b × (sqrt{a^2 - left(frac{b}{2}right)^2})

Solving for b

Rearrange the area equation to isolate b:

2A b × (sqrt{a^2 - left(frac{b}{2}right)^2})

Square both sides to eliminate the square root:

2A2 b2 × (a^2 - left(frac{b}{2}right)^2)

This simplifies to:

4A2 a2b2 - (frac{b^4}{4})

Rearrange the equation:

4A2 ( var{a^2 - frac{b^2}{4}}) b2

Further simplification leads to a quartic equation in terms of b as follows:

b4 - 4ab2 - 4A2 0

Solving the Quartic Equation

To solve for b, we can use the quadratic equation by letting x b2:

x2 - 4ax - 4A2 0

Solve for x using the quadratic formula:

x (frac{4a pm sqrt{4a2 - 4cdot1cdot4A2}}{2cdot1})

x 2a (pm sqrt{16a2 - 64A2})

Since x b2, b can be found as:

b (sqrt{2a pm sqrt{16a2 - 64A2}})

Conclusion

The third side b of the isosceles triangle can be calculated using the above steps given the lengths of the equal sides a and the area A. Always check the validity of the solutions to ensure the triangle inequality holds for the resulting lengths.