a right rectangle and a right triangle prism have the same height. If the lenght of the side of the rectangle base is twice the width, explain how to determine the area of the triangular base in terms of the rectangle base length.
To determine the area of the triangular base in terms of the rectangle base length, we can follow a step-by-step approach.
Step 1: Understand the Problem
We have a right rectangle and a right triangular prism with the same height. The rectangle's base has a length that is twice its width. We need to find the area of the triangular base in terms of the rectangle's base length.
Step 2: Define Variables
Let the width of the rectangle's base be 'w' units.
Therefore, the length of the rectangle's base would be '2w' units since it is twice the width.
Let 'h' represent the height of both the rectangle and the triangular prism.
Step 3: Calculate the Area of the Rectangle's Base
The area of a rectangle is given by the formula: Area = length * width.
Substituting the values, we have:
Area of Rectangle's Base = (2w) * w = 2w^2 square units.
Step 4: Calculate the Area of the Triangular Base
Since the triangular prism shares the same height as the rectangle, the triangular base's height will also be 'h' units.
To determine the length of the triangular base (which is also the base of the rectangle), we can use the Pythagorean theorem.
In a right triangle, the hypotenuse squared is equal to the sum of the squares of the other two sides. We know that one side of the triangle is 'w', the other side is '2w', and the hypotenuse represents the length of the triangular base.
Using the Pythagorean theorem, we can write the equation:
w^2 + (2w)^2 = triangular base length^2.
Simplifying this equation:
w^2 + 4w^2 = triangular base length^2.
5w^2 = triangular base length^2.
Taking the square root of both sides:
√(5w^2) = triangular base length.
Triangular base length = √5w.
Finally, we can calculate the area of the triangular base using the formula for the area of a triangle: Area = (base * height) / 2.
Substituting the values, we get:
Area of Triangular Base = (√5w * h) / 2 = (√5wh) / 2.
Therefore, the area of the triangular base is (√5wh) / 2, where 'w' is the width of the rectangle's base and 'h' is the common height of both the rectangle and the triangular prism.