The Pythagorean Theorem and its Converse
Standard
G.SRT.8 Use the Pythagorean Theorem to solve right triangles in applied problems.
Goals for this section:
Students will know how to use the Pythagorean theorem and its converse. Students will understand that the Pythagorean Theorem can be used to find the lengths of the sides of a right triangle.
Pythagorean Theorem
If ΔABC is a right triangle, then a2 + b2 = c2
- a and b are legs. These segments form the right angle.
- c is a hypotenuse. This segment is always across from the right angle.
Pythagorean Triple
A Pythagorean Triple is a set of nonzero whole numbers a, b, and c that satisfy the equation .
You can multiply these sets of numbers by the same whole number and get another Pythagorean triple.
Remember, the biggest number is the hypotenuse since the hypotenuse is the longest segment!
Here are some common triples:
3, 4, 5
5, 12, 13
8, 15, 17
7, 24, 25
Converse of the Pythagorean Theorem
If a2 + b2 = c2, then ΔABC
Meaning, if the three sides of a triangle satisfy a2 + b2 = c2 then it is a right triangle. This is the converse, which means "going the other way". We'll generally have questions asking if given a set of three numbers, does it make a right triangle?
- Tip: The largest number given should be the hypotenuse!
Acute and Obtuse Triangles
These types of triangles do not satisfy the converse of the Pythagorean Theorem.
Acute
Acute means smaller than 90 degrees. An acute triangle has angles that are smaller than 90 degrees.
c2 < a2 + b2
Obtuse
Obtuse means larger than 90 degrees. An obtuse triangle has at least one angle that is over 90 degrees.
c2 > a2 + b2
