Bellringer Solve for x. 1. x=4 2. x=7 3. x=8 4. x=10

GEOMETRY: CHAPTER 8 RIGHT TRIANGLES AND TRIGONOMETRY Sect. 8-1 The Pythagorean Theorem

Review The last chapter we discussed the relationship of shapes. We used proportions and ratios to find these similarities. (angles and sides) Thus, the similarity in right triangles will help us with the Pythagorean Theorem.

Objective Essential Question To explore and understand the relationship of the Pythagorean Theorem. Also, how to use the converse Pythagorean Theorem to determine if it is a right triangle or not. What problems can the Pythagorean Theorem help solve? Such that, if the lengths of two sides of a right triangle are known, the length of the third side can be found by the Pythagorean theorem. Lesson Purpose

Real World Connection So Pythagoras went to Babylon and studied with the Chaldean stargazers. He went to Egypt and studied the lore of the priests at Memphis and Diospolis. In Egypt he studied with the people known as the "rope-stretchers". These were the engineers who built the pyramids. They held a very special secret in the form of a rope tied in a circle with 12 evenly spaced knots. It turns out that if the rope was pegged to the ground in the dimensions of , a right triangle would emerge instantly.

History This enabled them to lay the foundations for their buildings accurately. He traveled to all the known parts of the Mediterranean world. During his travels he came to the conclusion that the earth must be round. In history, he is given credit as the first person to spread this idea. The most famous discovery that Pythagoras made came from his fascination with the Egyptian rope-stretchers triangle.

Discovery If the triangle had a right angle (90°) and you made a square on each of the three sides, then the biggest square had the exact same area as the other two squares put together! Theorem Proof

Theorem 8-1 If ∆ABC is a right triangle…. Then… a²+b²=c² Pythagorean Theorem

Demonstration In this clip see how the water from the two smaller square fill up the big square.

Solve for x with Pythagorean theorem. Solution: Step one: set up equation 21²+72²=x² =x² √5625=√x² x=75 Example #1

Solution Solve for x with Pythagorean theorem 4²+x²=5² 16+x²= x²=25-16 x²=9 √x²=√9 x=9 Example #2

Solve for x with Pythagorean theorem Solution 14²+48²=x² =x² 2500=x² √2500=√x² x=50 Example #3

Definition: Examples Is a set of nonzero whole numbers a, b, c that make the equation a²+b²=c² true. Most common triples are on the right. 3,4,5 5,12,13 8,15,17 7,24,25 Pythagorean Triple

If a, b, and c satisfy the equation a² + b² = c², then a, b, and c are known as Pythagorean triples. Finding a Pythagorean Triple

What about reversing the method to find m and n? a = 2mn, b = m 2 - n 2 and c = m 2 + n 2 where n (<)is less than m? In other words, m and n should be relatively prime and if one is odd the other is even. Of course part of the similarity issue is taken care of since m even and n even implies a common factor of 2. Finding a Pythagorean Triple

How can we pick m and n, with n less than m, in order that we get a Unique Pythagorean Triple? Try using natural numbers that have a greatest common factor of one and opposite parity. Questions to ask?

Example 4a Example 4b. Determine whether 4, 5, 6 is a Pythagorean triple. 4, 5, and 6 is not a Pythagorean triple Determine whether 15, 8, and 17 is a Pythagorean triple. 15, 8, and 17 is a Pythagorean triple Example #4

Questions to ask? If you know the lengths of two sides of a right triangle, how can you find the unknown side length? Substitute the known side lengths into the Pythagorean theorem and solve for the third side length.

If… a²+b²=c² Then ∆ABC is a right triangle. The converse is used to determine if three given side lengths form a right triangle. Converse of the Pythagorean Theorem

Theorem 8-4-Triangle Inequality

Theorem 8-3- Triangle Inequality

Solution Problem: Let c represent the length of the longest side of the triangle. Check to see whether the side lengths satisfy the equation c² = a² + b². (√113)²= 7² + 8² 113 = = 113 ✔ The triangle is a right triangle. Decide whether the set of numbers can represent the side lengths of a triangle. If they can, classify the triangle as right, acute or obtuse. 7,8,√113 Example #5a

Problem Solution Decide whether the set of numbers can represent the side lengths of a triangle. If they can, classify the triangle as right, acute or obtuse. 15,36, and 4√95 c2 = a2 + b2. (4√95)2 = ∙ (√95)2 = ∙ 95 = ≠ 1521 ✔ The triangle is NOT a right triangle. it is acute. Example #5b

Problem Solution Decide whether the set of numbers can represent the side lengths of a triangle. If they can, classify the triangle as right, acute or obtuse a.38, 77, 86 c² ? a² + b² 86² ? 38² + 77² 7396 ? > 7373 Reason: Compare c² with a² + b² Substitute values Multiply c² is greater than a² + b² The triangle is obtuse Example #6a

Problem Solution Decide whether the set of numbers can represent the side lengths of a triangle. If they can, classify the triangle as right, acute or obtuse b. 10.5, 36.5, 37.5 c²? a² + b² 37.5² ? 10.5²+ 36.5² ? < Reason: Compare c² with a² + b² Substitute values Multiply c² is less than a² + b² The triangle is acute Example #6b

Essential Understanding Therefore, anytime there is a right triangle and two sides are known you can use the Pythagorean theorem to find the third. Also, you can use the converse of the Pythagorean theorem to find out if it is a right triangle. Summary

Ticket out Homework: What is the difference between how the Pythagorean Theorem and its converse are used? The Pythagorean theorem is used to determine the length of the third side of a right triangle given two lengths. The converse is used to determine if three given side length form a right triangle. Pg #’s 4-26 even Pg #’s 4-20 even Ticket Out and Homework