7B Pythagorean Theorem and Its Converse OBJECTIVES: To determine missing measures using the Pythagorean Theorem To determine right triangles using the Converse of the Pythagorean Theorem

Right Triangle Parts Longest side Opposite rt. angle

THEOREM THEOREM: Pythagorean Theorem NOTE: The Pythagorean Theorem is useful in finding missing lengths of sides in right triangles THEOREM THEOREM: 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. Right ∆  c2 = a2 + b2 hypotenuse leg leg

Using the Pythagorean Theorem EXAMPLE 1: Finding the Length of a Hypotenuse Given a right triangle with legs of lengths 5 cm and 12 cm, find the length of the hypotenuse.

Using the Pythagorean Theorem EXAMPLE 2: Finding the Length of a Leg Given a right triangle with hypotenuse of length 14 cm and leg of length 7 cm, find the length of the remaining leg.

Using the Pythagorean Theorem Find the area of the triangle at the left to the nearest tenth of a square meter. Recall: In an isosceles triangle, the height is the median is the angle bisector.

THEOREM THEOREM: Converse of the Pythagorean Theorem NOTE: The Converse of the Pythagorean Theorem is useful in determining right triangles. THEOREM THEOREM: Converse of the Pythagorean Theorem 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 the triangle is a right triangle. c2 = a2 + b2  right triangle

The Pythagorean Theorem and Its Converse can be written as the following bi-conditional statement: Right ∆ c2 = a2 + b2

Using the Converse of the Pythagorean Theorem: EXAMPLE 4: Determining Right Triangles The triangles below appear to be right triangles. Determine whether they are right triangles. 8 7 4√95 15 √113 36

THEOREMS c2 < a2 + b2  acute triangle THEOREMS TO DETERMINE ACUTE OR OBTUSE TRIANGLES If the square of the length of the longest side of a triangle is ____________ the sum of the squares of the lengths of the other two sides, then the triangle is an __________ triangle. c2 < a2 + b2  acute triangle c2 > a2 + b2  obtuse triangle

EXAMPLE 5: Classifying Triangles Determine if a triangle can be formed given the following lengths of sides. If they can, classify the triangle as right, acute, or obtuse. a. 38 cm, 77cm, 86cm b. 10.5cm, 36.5cm, 37.5cm

To summarize: Pythagorean Theorem and Its Converse Right ∆  __________________ c2 = a2 + b2  ____________ Classifying Right Triangles c2 < a2 + b2  ____________ c2 > a2 + b2  ____________

Final Checks for Understanding State the Pythagorean Theorem in your own words. Which equations are true for ∆ PQR? a. r2= p2 + q2 b. q2= p2 + r2 c. p2= r2 - q2 d. r2= (p + q)2 Q r p P q R e. p2= q2 + r2

Final Checks for Understanding 3. State the Converse of the Pythagorean Theorem in your own words. 4. Match the lengths of the sides with the appropriate description. 5. 2, 10, 11 6. 13, 5, 7 7. 5, 11, 6 8. 6, 8, 10 A. right ∆ B. acute ∆ C. obtuse right ∆ D. not a ∆

HOMEWORK ASSIGNMENT: Pythagorean Theorem and Its Converse WS, plus textbook:_______________________