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Let’s prove the Pythagorean T heorem

Let’s prove the Pythagorean T heorem. Copy and study the diagram. Let’s establish some angle relationship. m  1+ m  2 = 90° (1) Acute angles in the right triangle ADC are complementary .

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Let’s prove the Pythagorean T heorem

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  1. Let’s prove the Pythagorean Theorem

  2. Copy and study the diagram

  3. Let’s establish some angle relationship m1+ m2 =90° (1) Acute angles in the right triangle ADC are complementary. m3+ m4 =90° (2) Acute angles in the right triangle CDB are complementary. m2+ m3 =90° (3) Complementary angles that form angle ACB m1+ m4 =90° (4) Acute angles in the right triangle ACB are complementary.

  4. Recall some properties of equality m1+ m2= m2+ m3Transitive property of equality for (1) and (3) m1= m3 (5) Subtraction property of equality m3+ m4 = m2+ m3 Transitive property of equality for (1) and (4) m4= m2(6) Subtraction property of equality

  5. Identify similar triangles ΔADCΔCDB ΔACB by AA similarity

  6. Write proportions for the corresponding sides • Consider ΔADCΔACB Corresponding sides are proportional (7) Cross productproperty of proportion

  7. Write proportions for the corresponding sides (continued) • Consider ΔCDBΔACB Corresponding sides are proportional (8) Cross product property of proportion

  8. Recall some more algebraic properties Distributive Property =Segment addition = The Pythagorean Theorem

  9. Extension: The geometric Mean Let’s revisit some of the proportions and equations that we used earlier. The values of b and are called the geometric means of the proportions. Solve the proportions for b and respectively.

  10. Extension: The geometric Mean In a right triangle each leg is the geometric mean between the hypotenuse and the segment of the hypotenuse adjacent to that leg.

  11. Application Adam, Ben, and Christopher are three friends. Their houses are located at the vertices of the right triangle in a rural area. Adam and Ben live 420 yards apart, and their houses are connected by a paved road . The distance from Adam’s house to Christopher’s is 336 yards, and the distance from Ben’s house to Christopher’s is 252 yards. Christopher doesn’t have access to the paved road so he wants to construct a road from his house to the road that connects his friends’ houses. What is the shortest possible length for his road?

  12. 5-minute check: Proving the Altitude and Segments of Hypotenuse Relationship Prove that in the right triangle the altitude drawn to the hypotenuse is the geometric mean between the segments of the hypotenuse. Hint. Apply the method used in the proof of the Pythagorean Theorem to write the proportions that include the length of (marked by h) on the diagram.

  13. Solution to the 5-minute check ΔADCΔCDB by AA similarity Cross Product property of proportion

  14. Challenge Prove that in the right triangle below h= Hint. Apply the idea that the altitude to the hypotenuse is the geometric mean between the segments of the hypotenuse.

  15. Answer to the Challenge Problem Recall the following relationships: Solving the first two equations for the lengths of the segments of the hypotenuse yields and . Substitute the new equations in the = Taking the square root of the last equation yields h=

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