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The Concept of Congruence Module two

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1 The Concept of Congruence Module two
Grade 8 The Concept of Congruence Module two

2 The Pythagorean Theorem
TOPIC D The Pythagorean Theorem

3 Informal proof of the Pythagorean theorem
8.2 NYS COMMON CORE MATHEMATICS CURRICULUM Informal proof of the Pythagorean theorem Lesson fifteen

4 Lesson fifteen 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM Concept Development Pythagorean Theorem: If the lengths of the legs of a right triangle are 𝑎 and 𝑏, and the length of the hypotenuse is 𝑐, then 𝑎2 + 𝑏2 = 𝑐2. Given a right triangle ABC with 𝐶 being the vertex of the right angle, then the sides AC and BC are called the legs of ∆ABC, and AB is called the hypotenuse of ∆ABC.

5 Basic Geometry Facts 1. Congruent triangles have equal areas.
Lesson fifteen 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM Basic Geometry Facts 1. Congruent triangles have equal areas. 2. All corresponding parts of congruent triangles are congruent. 3. The triangle sum theorem. (∠ sum of △) a. For right triangles, the two angles that are not the right angle have a sum of 90°. (∠ sum of rt. △)

6 Basic Geometry Facts Lesson fifteen
8.2 NYS COMMON CORE MATHEMATICS CURRICULUM Basic Geometry Facts

7 Determine the length of the hypotenuse of the right triangle.
Lesson fifteen 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM EXAMPLE TWO Determine the length of the hypotenuse of the right triangle.

8 Determine the length of the hypotenuse of the right triangle.
Lesson fifteen 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM EXERCISES 1-3 Determine the length of the hypotenuse of the right triangle.

9 Determine the length of the hypotenuse of the right triangle.
Lesson fifteen 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM EXERCISES 4-5 Determine the length of the hypotenuse of the right triangle.

10 Lesson Summary Lesson fifteen NYS COMMON CORE MATHEMATICS CURRICULUM
8.2 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson Summary

11 Applications of the Pythagorean theorem
8.2 NYS COMMON CORE MATHEMATICS CURRICULUM Applications of the Pythagorean theorem Lesson sixteen

12 Lesson sixteen 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM EXAMPLE ONE

13 Lesson sixteen 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM EXAMPLE TWO Suppose you have a ladder of length 13 feet. Suppose that to make it sturdy enough to climb, you must place the ladder exactly 5 feet from the wall of a building. You need to post a banner on the building 10 feet above the ground. Is the ladder long enough for you to reach the location you need to post the banner?

14 Lesson sixteen 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM EXAMPLE THREE Given a right triangle with a hypotenuse of length 15 units and a leg of length 9, what is the length of the other leg?

15 Lesson sixteen 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM EXERCISE 1

16 Lesson sixteen 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM EXERCISE 2

17 Lesson sixteen 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM EXAMPLE FOUR

18 Lesson sixteen 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM EXAMPLE FIVE Given a rectangle with side lengths of 8 cm and 2 cm, as shown, what is the length of the diagonal?

19 Find the length of the segment AB, if possible.
Lesson sixteen 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM EXERCISE 3 Find the length of the segment AB, if possible.

20 Lesson sixteen 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM EXERCISE 4 4. Given a rectangle with dimensions 5 cm and 10 cm, as shown, find the length of the diagonal, if possible.

21 Lesson sixteen 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM EXERCISE 5 & 6 A right triangle has a hypotenuse of length 13 in. and a leg with length 4 in. What is the length of the other leg? Find the length of 𝑏 in the right triangle below, if possible.

22 Lesson Summary Lesson sixteen NYS COMMON CORE MATHEMATICS CURRICULUM
8.2 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson Summary

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