The lengths of three sides of a triangle are 4, 6, and 10. Is this a right triangle?
There are many different proofs of the Pythagorean Theorem. The following picture leads to one of those proofs.
In order to prove the Pythagorean Theorem, find the area of the interior square in two ways. First, find the area directly:
Next, find the area as the difference between the area of the large square and the area of the triangles.
Since you are referring to the same square each time, those two areas must be equal.
Use algebra to simplify.
With the Pythagorean Theorem and its converse, you can solve many types of problems. You can:
- Find the missing side of a right triangle when you know the other two sides.
- Determine whether a triangle is right, acute, or obtuse.
- Find the distance between two points.
Finding the Length of the Hypotenuse
The two legs of a right triangle have lengths 3 and 4. What is the length of the hypotenuse?
Because length must be positive, the hypotenuse has a length of 5. Side lengths of 3, 4 and 5 are common in geometry. You should remember that they are the lengths of a right triangle. Triples of whole numbers that satisfy the Pythagorean Theorem are called Pythagorean triples. “3, 4, 5” is an example of a Pythagorean triple.
A triangle has side lengths of 4, 8 and 9. What type of triangle is this?
Deriving the Distance Formula
Earlier, you were asked if the length of the three sides given is a right triangle.
Will a multiple of a Pythagorean triple always also be a Pythagorean triple? For example, “6, 8, 10” is a multiple of “3, 4, 5”. Is “6, 8, 10” a Pythagorean triple? Is any multiple of “3, 4, 5” (or any other Pythagorean triple) also a Pythagorean triple?
Use the Pythagorean Theorem or the distance formula.
The length of one leg of a triangle is 5 and the length of the hypotenuse is 8. What is the length of the other leg?
Use the Pythagorean Theorem.
Three side lengths for triangles are given. Determine whether or not each triangle is right, acute, or obtuse.
6. 2, 5, 6
7. 4, 7, 8
8. 6, 8, 10
9. 6, 9, 10
Find the distance between each pair of points.
14. Find two more Pythagorean triples that are not multiples of “3, 4, 5”.
To see the Review answers, open this PDF file and look for section 1.7.