Luke is building a bike ramp. The ramp is going to be 3 feet tall on the high end and is 6 feet long. How long is the base of the ramp?

Steps to Solve

1. Identify the known values

In this problem, we have the height of the triangle ($h$) and the length of the hypotenuse ($r$). The height is given as 3 feet, and the length of the ramp (hypotenuse) is 6 feet.

• $h = 3 \text{ feet}$
• $r = 6 \text{ feet}$

2. Apply the Pythagorean theorem

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. This can be written as:

$$ r^2 = h^2 + b^2 $$

where $b$ is the length of the base we want to find.

3. Substitute the known values into the equation

Now we substitute $r$ and $h$ into the equation:

$$ 6^2 = 3^2 + b^2 $$

Simplifying gives:

$$ 36 = 9 + b^2 $$

4. Solve for the base

To find $b^2$, we rearrange the equation:

$$ b^2 = 36 - 9 $$

which simplifies to:

$$ b^2 = 27 $$

Now, take the square root of both sides to find $b$:

$$ b = \sqrt{27} = 3\sqrt{3} \text{ feet} $$