Let’s prove the Pythagorean Theorem

Copy and study the diagram

Let’s establish some angle relationshipm1+ m2 = 90° (1) Acute angles in the right triangle ADC are

complementary.

m3+ m4 = 90° (2) Acute angles in the right triangle CDB are complementary.

m2+ m3 = 90° (3) Complementary angles that form angle

ACB

m1+ m4 = 90° (4) Acute angles in the right triangle ACB are complementary.

Recall some properties of equality

m1+ m2 = m2+ m3 Transitive property of equality for

(1) and (3)

m1= m3 (5) Subtraction property of equality

m3+ m4 = m2+ m3 Transitive property of equality for

(1) and (4)

m4= m2 (6) Subtraction property of equality

Identify similar triangles

ΔADC ΔCDB ΔACB by AA similarity

Write proportions for the corresponding sides

• Consider ΔADC ΔACB

Corresponding sides are proportional

(7) Cross product property of proportion

Write proportions for the corresponding sides (continued)• Consider ΔCDB ΔACBCorresponding sides are proportional

(8) Cross product property of proportion

Recall some more algebraic properties

Distributive Property = Segment addition = The Pythagorean Theorem

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.

𝑎𝑐

𝑎=𝑎𝑐

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.

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?

5-minute check: Proving the Altitude and Segments of Hypotenuse RelationshipProve 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.

Solution to the 5-minute check

ΔADC ΔCDB by AA similarity

Cross Product property of proportion

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.

Answer to the Challenge ProblemRecall 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=