The Converse Of The Pythagorean Theorem Section 8-1.

The Converse Of The Pythagorean Theorem

The Converse Of The Pythagorean TheoremSection 8-1Objectives:Use the Converse of the Pythagorean Theorem.Use side lengths to classify triangles by their angles.Converse of the Pythagorean TheoremConverse of the Pythagorean TheoremIf the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then it is a right triangle.Example:

7.2 Converse of the Pythagorean Theorem3ANSWERIt is true that c2 = a2 + b2. So, ABC is a right triangle.Is ABC a right triangle?

SOLUTIONLet c represent the length of the longest side of the triangle. Check to see whether the side lengths satisfy the equation c2 = a2 + b2.400 = 400Simplify.Compare c2 with a2 + b2.c2 a2 + b2?==Multiply.400 144 + 256?==Substitute 20 for c, 12 for a, and 16 for b.202 122 + 162?==Example 14Determine whether 9, 12, and 15 are the sides of a right triangle. Since the measure of the longest side is 15, 15 must be c. Let a and b be 9 and 12.Pythagorean TheoremSimplify.Add.

Example 2a:Answer: These segments form the sides of a right triangle since they satisfy the Pythagorean Theorem. 5Determine whether 21, 42, and 54 are the sides of a right triangle. Pythagorean TheoremSimplify.Add.

Answer: Since , segments with these measures cannot form a right triangle.

Example 2b:6Pythagorean TheoremSimplify.Add.Determine whether 43, 4, and 8 are the sides of a right triangle.

Answer: Since 64 = 64, segments with these measures form a right triangle.

Example 2c:7Answer: The segments form the sides of a right triangle.Answer: The segments do not form the sides of a right triangle.Answer: The segments form the sides of a right triangle.Your Turn:Determine whether each set of measures are the sides of a right triangle. a. 6, 8, 10

b. 5, 8, 9

c.

8More PracticeWhich of the following is a right triangle?

272(729)202+152(625)202(400)152+122(369)302(900)182+252(949)652(4225)=602+252(4225)NONONOYESClassifying trianglesClassifying TrianglesUsing the Converse of the Pythagorean Theorem we can classify a triangle as acute, right, or obtuse by its side lengths.Obtuse1 angle is obtuse (measure > 90)Right1 angle is right(measure = 90)Acuteall 3 angles are acute (measure < 90)Classifying TrianglesAcute Triangle If the square of the length of the longest side of a triangle is less than the sum of the squares of the lengths of the other two sides, then it is an acute triangle.Example:

Classifying TrianglesRight Triangle If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then it is a right triangle.Example:

Then triangleABC is rightClassifying TrianglesObtuse Triangle If the square of the length of the longest side of a triangle is greater than the sum of the squares of the lengths of the other two sides, then it is an obtuse triangle.Example:

Summary

This is the Converse of the Pythagorean TheoremANSWERBecause c2 < a2 + b2, the triangle is acute. Show that the triangle is an acute triangle.SOLUTIONCompare the side lengths.35 < 41Simplify.

Compare c2 with a2 + b2.c2 a2 + b2?==Multiply.35 16 + 25?==Substitute for c, 4 for a, and 5 for b.352 42 + 52

?==35Example 316ANSWERBecause c2 > a2 + b2, the triangle is obtuse.Show that the triangle is an obtuse triangle.SOLUTIONCompare the side lengths.225 > 208Simplify.

c2 a2 + b2Compare c2 with a2 + b2.?==(15)2 82 + 122Substitute 15 for c, 8 for a, and 12 for b.?==225 64 + 144Multiply.?==Example 417

ANSWERBecause c2 > a2 + b2, the triangle is obtuse.Classify the triangle as acute, right, or obtuse.SOLUTIONCompare the square of the length of the longest side with the sum of the squares of the lengths of the two shorter sides.64 > 71Simplify.64 25 + 36Multiply.?==82 52 + 62Substitute 8 for c, 5 for a, and 6 for b.?==c2 a2 + b2Compare c2 with a2 + b2.?==Example 518Classify the triangle with the given side lengths as acute, right, or obtuse.a.4, 6, 7b.12, 35, 3749 < 521369 = 1369The triangle is acute.The triangle is right.SOLUTIONa.c2 a2 + b2?==b.c2 a2 + b2?==372 122 + 352?==72 42 + 62?==1369 144 + 1225?==49 16 + 36?==Example 619ANSWERobtuse; 62 > 22 + 52 36 >29ANSWERright; 172 = 82 + 152 289 = 289ANSWERacute; 72 < 72 + 72 49 < 98Classify the triangle as acute, right, or obtuse. Explain.1.

2.

3.

Your Turn:ANSWERacuteANSWERrightANSWERobtuseUse the side lengths to classify the triangle as acute, right, or obtuse.4.7, 24, 245.7, 24, 256.7, 24, 26Your Turn:Example 7 Classify the triangle with lengths 9, 12, and 15 as acute, right, or obtuse. Justify your answer.c2= a2 + b2Compare c2 and a2 + b2.?152= 122 + 92Substitution?225= 225Simplify and compare.Answer:Since c2 = a2 + b2, the triangle is right.Example 8Classify the triangle with lengths 10, 11, and 13 as acute, right, or obtuse. Justify your answer.c2= a2 + b2Compare c2 and a2 + b2.?132= 112 + 102Substitution?169< 221Simplify and compare.Answer:Since c2 < a2 + b2, the triangle is acute.Your Turn:A.acuteB.obtuseC.rightA. Classify the triangle with lengths 7, 8, and 14 as acute, right, or obtuse. Your Turn:A.acuteB.obtuseC.rightB. Classify the triangle with lengths 26, 22, and 33 as acute, right, or obtuse. Joke TimeWhat is black, white and red all over? You fold it!A newspaper.

How can you tell if there is an elephant in your fridge? You can't close the door.

What did one wall say to the other? "I'll meet you in the corner."