SAT Prep Question #11
Question: The figure shows a right triangle labeled ABC. One side is labeled 4. The area of the triangle is 10. What is the length of side CB?
This problem is asking you to find the length of the hypotenuse of the displayed triangle. Since it's a right triangle, you can use the Pythagorean theorem to determine the hypotenuse. However, this requires knowing the lengths of the other two sides of the triangle.
The problem provides you with the length of one side as well as the area of the triangle. Using the information provided, you can find the missing length and then plug it into the Pythagorean theorem to find the length of the hypotenuse.
In order to solve the problem, you need to know that the area of a triangle is equal to one half its base multiplied by its height (In a right triangle, the height is just the length of the other leg):
A=1/2(b)(h)
You also need to know that the Pythagorean theorem states that if you square the lengths of both sides of a right triangle, they will equal the square of the hypotenuse:
a²+b²=c²
Solve the Problem
1. First you need to find the length of the other side of the triangle. The base (length AB) and the area are already provided, so you can use them in the area formula to determine the other side.
Plug the given values into the area formula:
10=1/2(4)(h)
2. Solve the equation to find the height. The height is line segment AC, which is the missing side.
10=2(AC)
Divide both sides by 2. This gives you AC=5.
3. Now that you know the missing side, you can plug it into the Pythagorean theorem along with the side length that was provided. For this triangle, the equation would look like:
(AC)²+(AB)²=(CB)²
Plug in the values to get:
(5)²+(4)²=(CB)²
4. Now solve for CB, which is the hypotenuse of the triangle.
25+16=(CB)²
41=(CB)²
You will need to take the square root of both sides to find length CB. This gives you √41=CB. This is the simplest way to write the answer.
The hypotenuse is √41.