How to find the third side of an isosceles triangle?
Understand the Problem
The question is asking for a method to calculate the length of the third side of an isosceles triangle, given the lengths of the other two sides (which are equal). This likely involves using basic geometric principles or the Pythagorean theorem when dealing with the height of the triangle.
Answer
The length of the third side is given by $b = 2\sqrt{a^2  h^2}$ if the height $h$ is known.
Answer for screen readers
The length of the third side (base) of the isosceles triangle can be calculated using: $$ b = 2\sqrt{a^2  h^2} $$ if the height $h$ is known.
Steps to Solve

Identify the lengths of the equal sides First, denote the lengths of the two equal sides of the isosceles triangle as $a$. The length of the base, which is the third side, will be denoted as $b$.

Use the formula for isosceles triangles To find the length of the base $b$, we can utilize the properties of the isosceles triangle. We can drop a perpendicular from the vertex opposite the base to the midpoint of the base. This creates two right triangles.

Apply the Pythagorean theorem Let $h$ be the height of the triangle. Using the Pythagorean theorem in one of the right triangles, we have: $$ a^2 = \left(\frac{b}{2}\right)^2 + h^2 $$

Express the base in terms of the height and side lengths Rearranging the Pythagorean theorem gives us: $$ h^2 = a^2  \left(\frac{b}{2}\right)^2 $$ To find $b$, we can set $h$ directly based on the specific dimensions or angles given, but we need additional information such as the height or an angle to find $b$ directly.

Solve for base length (if height is known) If we have a specific height $h$, we rearrange the equation for $b$: $$ \left(\frac{b}{2}\right)^2 = a^2  h^2 $$ then, $$ b = 2\sqrt{a^2  h^2} $$

Conclusion Depending on whether additional information is given (like height or angles), apply the identities and substitutions to solve for $b$.
The length of the third side (base) of the isosceles triangle can be calculated using: $$ b = 2\sqrt{a^2  h^2} $$ if the height $h$ is known.
More Information
The isosceles triangle has two sides of equal length, which simplifies calculations. The height helps separate the triangle into two rightangled triangles, allowing the use of the Pythagorean theorem. Knowing either the height or angle can help calculate the base.
Tips
 Forgetting to square the height when applying the Pythagorean theorem.
 Confusing the base with one of the equal sides.
 Incorrectly assuming angles when calculating base length without height information.