Why don’t we see mathematicians at the beach very often? They don’t need the sun because they have sine and cosine to give them a tan! If you don’t get this joke, you will after reading this study guide. Right triangles play a large role in trigonometry. The Pythagorean theorem and many trigonometric ratios depend on the properties of right triangles.
Triangle: A closed figure made by three line segments intersecting at their endpoints.
Vertex (plural, vertices): The point where the segments intersect.
Right Triangle: A triangle that contains one right angle.
Acute Triangle: A triangle with an acute angle.
Obtuse Triangle: A triangle with an obtuse angle.
Hypotenuse: Longest leg in a right triangle, opposite of the 90° angle.
Pythagorean Triple: Three integers that satisfy the Pythagorean Theorem.
The sides and angles of triangles are generally named as shown in the picture below.
Pythagorean Theorem: For a right triangle with legs of lengths a and b and hypotenuse of length c, then:
\(\color{blue}{a}\color{black}{^2 + }\color{blue}{b}\color{black}{^2 = }\color{red}{c}\color{black}{^2}\)
Given the lengths of two sides of a right triangle, the Pythagorean Theorem can be used to find the third side length of the triangle.
A Pythagorean triple are three positive integers that make the Pythagorean Theorem true.
One example of a Pythagorean triple is the lengths 3, 4,and 5. Other Pythagorean triples can result by multiplying each integer in the triple by the same factor. For example, 3×2, 4×2, 5×2 = 6, 8, 10, which is another Pythagorean triple. This gives us an infinite number of Pythagorean triples. Some of the frequently used Pythagorean triples are:
The largest length must be the hypotenuse - if a triangle has two legs of lengths 3 and 5, the hypotenuse is not 4!
Tip: It is helpful to memorize these frequently used triples.
Converse of the Pythagorean Theorem: If the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.
The theorem can be extended to determine if a triangle is obtuse or acute.
Theorem: If the sum of the squares of the two shorter sides is greater than the square of the longest side, then the triangle is acute.
Theorem: If the sum of the squares of the two shorter sides is less than the square of the longest side, then the tri-angle is obtuse.
Using Side Lengths to Classify Triangles by Angles
If c² = a² + b², then m ∠C = 90º
and △ABC is a right triangle.
If c² < a² + b², then m ∠C < 90º
and △ABC is an acute triangle.
If c² > a² + b², then m ∠C > 90º
and △ABC is an obtuse triangle.
The Pythagorean Theorem can also be used to prove the distance formula, which is:
\(\sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}\)
1. Start with a line segment of unknown length (d) whose endpoints are
at \((x_1, y_1)\) and \((x_2, y_2)\)
2. Draw the horizontal and vertical lengths to create a right triangle using the line segment as the hypotenuse.