Isosceles Triangle Perimeter, Base and Sides

An isosceles triangle is a triangle in which two sides are equal in length. The perimeter of an isosceles triangle is the sum of the lengths of all three sides. In this problem, we are given the perimeter of an isosceles triangle and the length of the base.

Given
- Perimeter of isosceles triangle = 42 cm
- Base = 3/2 times the length of each equal side

Finding the Length of the Sides

Let x be the length of each equal side. Then, the length of the base is 3/2 times x. We can write an equation for the perimeter of the triangle in terms of x:

Perimeter = x + x + 3/2x = 42

Simplifying the equation:

2.5x = 42

x = 16.8

Therefore, the length of each equal side is 16.8 cm, and the length of the base is 3/2 times 16.8, or 25.2 cm.

Finding the Height

To find the height of the triangle, we need to use the Pythagorean theorem. Let h be the height of the triangle. Then, we can write:

h^2 = (16.8/2)^2 - (25.2/2)^2

Simplifying:

h^2 = 56.16

h = 7.5

Therefore, the height of the triangle is 7.5 cm.

Finding the Area

To find the area of the triangle, we can use the formula:

Area = 1/2 * base * height

Substituting the values we found:

Area = 1/2 * 25.2 * 7.5 = 94.5 cm^2

Therefore, the area of the isosceles triangle is 94.5 cm^2.

Conclusion

In this problem, we found the length of the sides, height, and area of an isosceles triangle given its perimeter and the length of its base. We used the Pythagorean theorem to find the height and the formula for the area of a triangle to find the area.