\(\newcommand{\substitute}[1]{{\color{blue}{{#1}}}}\newcommand{\amp}{&}\)
A meditation garden is in the shape of a right triangle, with one leg that is feet long. The length of the hypotenuse is foot longer than twice the length of the other leg. Write and solve a quadratic equation by factoring to find the lengths of the hypotenuse and the other leg.
The length of the hypotenuse is
feet and the length of the other leg is
feet.
It may help to draw a picture, like the one in the previous example. Since the garden is a right triangle, we know the Pythagorean Theorem must hold. So, we can substitute in for in the Pythagorean Theorem, and for which represents the length of the other leg. Then, we can substitute in for which represents the length of the hypotenuse.
Plugging these into the formula will give us:
Simplify both sides:
As you can see, this is a quadratic equation. So, we will solve this equation by factoring. First, we’ll need to get it into standard form and then we can use the Zero Product Property.
Of course, a negative length doesn’t make sense, so we’ll ignore the negative solution. Thus, the length of the other leg is feet long, which means the hypotenuse is one foot more than feet long. Therefore, the length of the hypothenuse is feet.