Let us assume that we have an isosceles triangle with sides a, a, and b (where b is the base), and the angle between the two equal sides is θ. To find the length of the sides of this triangle, we can use the following formulas:

• The length of the base (b) can be found using the Pythagorean theorem: b = √(a^2 – (a/2)^2) = √(3/4)a.

• The length of the equal sides (a) can be found using the sine formula: a = (b/2) / sin(θ/2).

Let’s work through an example to see how these formulas can be used to find the length of the sides of an isosceles triangle.

Example:

Find the length of the sides of an isosceles triangle if the base is 8 cm long and the angle between the two equal sides is 60 degrees.

Solution:

First, we can use the Pythagorean theorem to find the length of one of the equal sides:

b = √(3/4)a 8 = √(3/4)a a = 8 / √(3/4) a ≈ 9.237 cm

Next, we can use the sine formula to find the angle between the equal sides:

sin(θ/2) = (b/2) / a sin(θ/2) = 4 / 9.237 θ/2 ≈ 26.565 degrees θ ≈ 53.13 degrees

Therefore, the length of the sides of the isosceles triangle are approximately 9.237 cm and the angle between the two equal sides is approximately 53.13 degrees.

In conclusion, the length of the sides of an isosceles triangle can be found using the Pythagorean theorem and the sine formula. These formulas can be used to solve various problems involving isosceles triangles, such as finding the area or perimeter of the triangle.