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How to Calculate the Height of an Isosceles Triangle Using a Formula

Definition of an Isosceles Triangle

An isosceles triangle is a type of triangle with two equal sides and two equal angles. The base of an isosceles triangle is the side that is not equal to the other two sides. The height of an isosceles triangle is the distance from the vertex (the point where the two equal sides meet) to the base.

Formula for the Height of an Isosceles Triangle

The formula for the height of an isosceles triangle is: ``` h = √(s^2 - (b/2)^2) ``` where: * h is the height of the triangle * s is the length of one of the equal sides * b is the length of the base

Derivation of the Formula

The formula for the height of an isosceles triangle can be derived using the Pythagorean theorem. Let's draw an isosceles triangle with vertex A, base BC, and equal sides AB and AC. Draw a line from vertex A perpendicular to the base BC, intersecting BC at point D. This line segment, AD, is the height of the triangle. Using the Pythagorean theorem on triangle ABD, we get: ``` AD^2 = AB^2 - BD^2 ``` Since BD = BC/2, we have: ``` AD^2 = AB^2 - (BC/2)^2 ``` Solving for AD, we get: ``` AD = √(AB^2 - (BC/2)^2) ``` But AD is the height of the triangle, so we have: ``` h = √(AB^2 - (BC/2)^2) ```

Example

Let's say we have an isosceles triangle with equal sides of length 10 cm and a base of length 8 cm. We can use the formula to calculate the height of the triangle: ``` h = √(10^2 - (8/2)^2) ``` ``` h = √(100 - 16) ``` ``` h = √84 ``` ``` h = 9.16 cm ``` Therefore, the height of the isosceles triangle is 9.16 cm.

Conclusion

The formula for the height of an isosceles triangle is a useful tool for calculating the height of any isosceles triangle. This formula can be derived using the Pythagorean theorem and is a valuable tool for understanding the properties of isosceles triangles.