a rectangle with side of 6 cm and 5 cm has the same area as an isosceles triangle with base length 12 cm find the altitude to the base of the triangle and the sides of the triangle
So we need a triangle with area of 30 cm^2
let the length of each of the equal sides of the isosceles triangle be x, and the height be h
h^2 + 6^2 = x^2 **
but we also know that the area of the triangle is
(1/2)(12)h = 30
6h = 30
h = 5
then in **
25 + 36 = x^2
x^2 = 61
x = √61
altitude to the base of the triangle is 5 cm
the length of each of the equal sides is √61 cm
To find the altitude of the isosceles triangle, we need to start by finding the area of both the rectangle and the triangle.
The formula for the area of a rectangle is: Area = length × width
Given that the sides of the rectangle are 6 cm and 5 cm, the area can be calculated as: Area of rectangle = 6 cm × 5 cm
Next, let's calculate the area of the isosceles triangle. The formula for the area of a triangle is: Area = (base × altitude) / 2
We are given the base length of the triangle as 12 cm. So, we need to solve the equation:
Area of rectangle = Area of triangle
(6 cm × 5 cm) = (12 cm × altitude) / 2
Simplifying the equation further, we get:
30 cm² = 6 cm × altitude
Divide both sides of the equation by 6 cm:
5 cm = altitude
Hence, the altitude of the isosceles triangle is 5 cm.
To calculate the lengths of the other two sides of the triangle, we can use the Pythagorean theorem. In an isosceles triangle, the two equal sides are the legs, and the base is the hypotenuse.
The Pythagorean theorem states that in a right triangle, the sum of the squares of the legs is always equal to the square of the hypotenuse.
Let's denote the length of the equal sides as x and the length of the base (hypotenuse) as 12 cm.
So, we have:
x² + x² = 12²
Simplifying the equation further:
2x² = 144
Divide both sides of the equation by 2:
x² = 72
Taking the square root of both sides:
x = √72
x ≈ 8.49 cm
Hence, the lengths of the two equal sides of the isosceles triangle are approximately 8.49 cm, and the altitude to the base is 5 cm.