Similar Triangles and Right Triangles

    The idea of similar triangles has been around for thousands of years and is present in Euclid's book The Elements. Similar triangles are important for working with triangles but the importance also lies in the fact that similar triangles allow us to define the trigonometric functions. In this topic we explain similar triangles and state the Pythagorean Theorem and its converse. Succinctly, the Pythagorean Theorem states: in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. The converse is also true: if the square of the length of the longest side is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.

Definition (Similar Triangles) Two triangles are similar when the three angles are equal to the corresponding three angles of the other triangle.

Proposition (Similar Triangles) The corresponding sides of similar triangles are proportional. If and are the corresponding sides of two similar triangles, then

Example (Similar Triangles) Find the side for the pair of similar triangles:

    Solution. We have and solving for we have Therefore,

Definition (Right Triangles) Any triangle with two perpendicular sides is called a right triangle.

Proposition (Pythagorean Theorem) In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

Example (Pythagorean Theorem) (a) Find the hypotenuse of the right triangle determined by and  
    Solution. We have and solving for we have
(b) Find the leg of the right triangle determined by and hypotenuse
    Solution. We have and solving for we have

Proposition (Converse of the Pythagorean Theorem) If the square of the length of the longest side is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.

Example (Converse of the Pythagorean Theorem) Determine if the sides of the triangle make a right triangle.
    Solution. Since is the longest side and, are equal, this is a right triangle.

Example (Special Triangles) The and   triangles are sometimes called special triangles because of their heavy use with the trigonometric functions.

Example (Similar and Right Triangles) (a)  Find the side for the pair of similar triangles:

    
    Solution. We have and solving for we have   

(a) Determine if the sides of the triangle 1.50, 2.80, and 3.18 make a right triangle.
    Solution. Since 3.18 is the longest side and, and are not equal, this is not a right triangle.

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