To find the altitude of an isosceles triangle, we need to use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.

Let’s say that we have an isosceles triangle with sides of length a, a, and b, where b is the base. We want to find the altitude h, which is the length of the perpendicular line segment from the vertex opposite the base to the base.

To do this, we draw a line from the vertex opposite the base to the midpoint of the base, creating two right triangles. The altitude h is the hypotenuse of one of the right triangles, and the two sides of the right triangle are half of the base (b/2) and the height from the vertex to the midpoint of the base (h/2).

Using the Pythagorean theorem, we have:

h^2 = (b/2)^2 + (h/2)^2 h^2 = b^2/4 + h^2/4 3h^2/4 = b^2/4 h^2 = (4/3)b^2/4 h = sqrt((4/3)b^2/4) h = (sqrt(3)/2)b

Therefore, the altitude of an isosceles triangle with sides of length a, a, and base b is (sqrt(3)/2)b.

Let’s take an example to understand it better:

Example: Find the altitude of an isosceles triangle with sides of length 8 cm and base of length 10 cm.

Solution: We know that the altitude of an isosceles triangle with sides of length a, a, and base b is (sqrt(3)/2)b.

Substituting a = 8 and b = 10, we get:

h = (sqrt(3)/2)(10) h = 5sqrt(3)

Therefore, the altitude of the given isosceles triangle is 5sqrt(3) cm.