1. © 2010 Pearson Education, Inc. All rights reserved Concepts of Measurement Chapter 13

2. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 13.3- 2 NCTM Standard: Measurement Recognizing that objects have attributes that are measurable is the first step in the study of measurement. Children in prekindergarten through grade 2 begin by comparing and ordering objects using language such as longer and shorter. Length should be the focus in this grade band, but weight, time, area, and volume should also be explored.

3. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 13.3- 3 NCTM Standard: Measurement In grades 3–5, students should learn about area more thoroughly, as well as perimeter, volume, temperature, and angle measure. In these grades, they learn that measurements can be computed using formulas and need not always be taken directly with a measuring tool. Middle-grade students build on these earlier measurement experiences by continuing their study of perimeter, area, and volume and by beginning to explore derived measurements, such as speed. (p. 44)

4. Slide 13.3- 4 Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 13-3The Pythagorean Theorem, Distance Formula and Equation of a Circle  Special Right Triangles  Converse of the Pythagorean Theorem  The Distance Formula: An Application of the Pythagorean Theorem  Using the Distance Formula to Develop the Equation of a Circle

5. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 13.3- 5 Parts of a Right Triangle

6. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 13.3- 6 Pythagorean Theorem Given a right triangle with legs a and b and hypotenuse c, c 2 = a 2 + b 2. If BC = 3 cm and AC = 4 cm, what is the length of AB? 5 cm

7. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 13.3- 7 Proof of the Pythagorean Theorem 1.2.

8. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 13.3- 8 Proof of the Pythagorean Theorem Start with the right triangle. Draw squares on each leg and on the hypotenuse. Cut the square on one leg into four sections as shown. Then arrange sections and square to fill in the square on the hypotenuse.

9. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 13.3- 9 Example 13-12 The size of a rectangular television screen is given as the length of the diagonal of the screen. If the length of the screen is 24 in. and the width is 18 in., what is the diagonal length? The diagonal is 30 inches long.

10. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 13.3- 10 Example 13-13 A pole, BD, 28 ft high, is perpendicular to the ground. Two wires, BC and BA, each 35 ft long, are attached to the top of the pole and to stakes A and C on the ground. If points A, D, and C are collinear, how far are the stakes A and C from each other?

11. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 13.3- 11 Example 13-14 How tall is the Great Pyramid of Cheops, a right regular square pyramid, if the base has a side 771 ft and the slant height (altitude of ) is 620 ft? The Great Pyramid is approximately 485.6 feet tall.

12. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 13.3- 12 Special Right Triangles 45 ° -45 ° -90 ° right triangle The length of the hypotenuse in a 45°-45°-90° (isosceles) right triangle is times the length of a leg.

13. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 13.3- 13 Special Right Triangles In a 30°-60°-90° right triangle, the length of the hypotenuse is twice the length of the leg opposite the 30° angle (the shorter leg). The leg opposite the 60° angle (the longer leg) is times the length of the shorter leg. 30°-60°-90° right triangle

14. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 13.3- 14 Converse of the Pythagorean Theorem If  ABC is a triangle with sides of lengths a, b, and c such that c 2 = a 2 + b 2, then  ABC is a right triangle with the right angle opposite the side of length c.

15. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 13.3- 15 Example 13-15 Determine if the following can be the lengths of the sides of a right triangle: a.51, 68, 85b.2, 3,c.3, 4, 7 yes no

16. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 13.3- 16 The Distance Formula: An Application of the Pythagorean Theorem

17. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 13.3- 17 The Distance Formula The distance between the points A(x 1, y 1 ) and B(x 2, y 2 ) is given by

18. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 13.3- 18 Show that A(7, 4), B( – 2, 1), and C(10, − 4) are the vertices of an isosceles triangle. Then show that  ABC is a right triangle. AB = AC, so the triangle is isosceles. Example 13-16

19. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 13.3- 19 Example 13-16 (continued)  ABC is a right triangle with hypotenuse BC.

20. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 13.3- 20 Determine whether the points A(0, 5), B(1, 2), and C(2, − 1) are collinear. Example 13-17 If they are not collinear, they would be the vertices of a triangle, and hence AB + BC would be greater than AC (triangle inequality). If AB + BC = AC, a triangle cannot be formed and the points are collinear.

21. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 13.3- 21 Example 13-17 (continued) Since AB + BC = AC, the points are collinear.

22. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 13.3- 22 Using the Distance Formula to Develop the Equation of a Circle From the distance formula, we have The equation of a circle with the center at the origin and radius r is

23. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 13.3- 23 Using the Distance Formula to Develop the Equation of a Circle From the distance formula, we have The equation of a circle with the center (h, k) and radius r is