Unit 5, Lesson 4.3 Proving the Pythagorean Theorem using Similarity
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1 Unit 5, Lesson 4.3 Proving the Pythagorean Theorem using Similarity Geometry includes many definitions and statements. Once a statement has been shown to be true, it is called a theorem. Theorems, like definitions, can be used to show other statements are true. One of the most well known theorems of geometry is the Pythagorean Theorem, which relates the length of the hypotenuse of a right triangle to the lengths of its legs. The Pythagorean Theorem has many applications and can be very helpful when solving real-world problems. There are several ways to prove the Pythagorean Theorem; one way is by using similar triangles and similarity statements. The Pythagorean Theorem is often used to find the lengths of the sides of a right triangle, a triangle that includes one 90 angle. The Pythagorean Theorem The sum of the squares of the lengths of the legs (a and b) of a right triangle is equal to the square of the length of the hypotenuse (c). a + b = c In the triangle diagram above, angle C is 90, as shown by the square. The longest side of the right triangle, c, is called the hypotenuse and is always located across from the right angle. The legs of the right triangle, a and b, are the two shorter sides. It is also true that if the sum of the squares of the measures of two sides of a triangle equals the square of the measure of the longest side, then the triangle is a right triangle. This is known as the converse of the Pythagorean Theorem. To prove the Pythagorean Theorem using similar triangles, you must first identify the similar triangles. In this example, there is only one triangle given. Begin by drawing the altitude, the segment from angle C that is perpendicular to the line containing the opposite side, c. Notice that by creating the altitude CD, you have created two smaller right triangles, BDC, within the larger given right triangle, CB. CB and DC are 90 and are therefore congruent. DC of DC is congruent to of CB because of the Reflexive Property of Congruence. and Unit5Lesson /4/017
2 ccording to the ngle-ngle () Similarity Statement, if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar; therefore, DC ~ CB. CB and BDC are 90 and are therefore congruent. B of BDC is congruent to B of CB Two angles in Similarity is transitive. Since BDC are congruent to two angles in CB DC because of the Reflexive Property of Congruence. ~ CB and BDC ; therefore, BDC ~ CB. ~ CB, then DC ~ BDC. Corresponding sides of similar triangles are proportional; therefore, c b c a and. b d a e Determining the cross products of each proportion leads to the Pythagorean Theorem. c b c a b d a e cd b ce a ce cd a b dd both equations. c ( e d) a b Factor. c(c ) a b e d c c a b because of segment addition. The converse of the Pythagorean Theorem can be useful when proving right triangles using similar triangles. Types of Proofs Paragraph proofs are statements written out in complete sentences in a logical order to show an argument. Flow proofs are a graphical method of presenting the logical steps used to show an argument. In a flow proof, the logical statements are written in boxes and the reason for each statement is written below the box. Unit5Lesson4.3 11/4/017
3 nother accepted form of proof is a two-column proof. Two-column proofs include numbered statements and corresponding reasons that show the argument in a logical order. Two-column proofs appear in the Guided Practice examples that follow. Common Errors/Misconceptions misidentifying the altitudes of triangles incorrectly simplifying expressions with square roots Example 1 Write a two-column proof to prove the Pythagorean Theorem using similar triangles. b c C a B 1. Draw in helpful information. Draw in the altitude from Label D as e. Label DB as f. C and label the point of intersection on B as D. b c C a B. Identify the similar triangles. It is often helpful to redraw the triangles. CD D b c D BC C CBD C a B Unit5Lesson /4/017
4 Example 1 (continued) 3. Create the two-column proof. Statements Reasons 1. BC with right C 1. Given. BC ~ CD. If the altitude is drawn to the hypotenuse of a right triangle, BC ~ CBD then the two triangles formed are similar to the original triangle and each other. c a c b 3. ; a f b e 3. Definition of similar triangles; corresponding sides are proportional. 4. cf a ; ce b 4. Multiplication Property of Equality 5. cf ce a b 5. ddition Property of Equality 6. c( f e) a b 6. Distributive Property of Equality 7. e f c 7. Segment ddition Postulate 8. c(c ) Example a b OR 8. Substitution Property c a b Find the length of the altitude, x, of BC. 1. Identify the similar triangles. BC is a right triangle. The altitude of triangles. BC is drawn from CB to the opposite side, creating two smaller similar BC ~ CD ~ CBD. Use corresponding sides to write a proportion containing x. shorter leg of CD longer leg of CD shorter leg of CBD longer leg of CBD x x x ( x ) 10 (18 ) Find the cross products. x 18 0 Simplify. x 180 Take the positive square root of each side. x 6 5 x Summarize your findings. The length of the altitude, x, of BC is 6 5 units, or approximately 13.4 units. Unit5Lesson /4/017
5 Example 3 Find the unknown values in the figure. 1. Identify the similar triangles. BC is a right triangle. The altitude of BC is drawn from right CB to the opposite side, creating two smaller similar triangles. BC ~ CD ~ CBD. Use corresponding sides to write a proportion containing c. hypotenuse of CD shorter leg of CD hypotenuse of BC shorter leg of BC c 6 8( 6 ) c ( 4.8 ) Find the cross products c Simplify. c 10 Solve for c. 3. Use corresponding sides to write a proportion containing e. longer leg of CD shorter leg of CD longer leg of BC shorter leg of BC e e( 6 ) 8( 4.8 ) Find the cross products. 6e 38.4 Simplify. e 6.4 Solve for e. 4. Use corresponding sides to write a proportion containing f. longer leg of CBD shorter leg of CBD longer leg of BC shorter leg of BC 4.8 f ( 6) 8( f ) Find the cross products f Simplify. f 3.6 Solve for f. 5. The Summarize your findings. The length of c is 10 units. The length of e is 6.4 units. The length of f is 3.6 units. Unit5Lesson /4/017
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