Chapter 14 Area Pythagorean Theorem and Volume Copyright

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Chapter 14 Area, Pythagorean Theorem, and Volume Copyright © 2013, 2010, and 2007, Pearson

Chapter 14 Area, Pythagorean Theorem, and Volume Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

14 -2 The Pythagorean Theorem, Distance Formula and Equation of a Circle § Special

14 -2 The 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 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Parts of a Right Triangle Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Parts of a Right Triangle Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Pythagorean Theorem Given a right triangle with legs a and b and hypotenuse c,

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 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 14 -7 b The size of a rectangular television screen is given as

Example 14 -7 b 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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 14 -8 A pole, BD, 28 ft high, is perpendicular to the ground.

Example 14 -8 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? Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 14 -9 How tall is the Great Pyramid of Cheops, a right regular

Example 14 -9 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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Special Right Triangles 45°-90° right triangle The length of the hypotenuse in a 45°-90°

Special Right Triangles 45°-90° right triangle The length of the hypotenuse in a 45°-90° (isosceles) right triangle is times the length of a leg. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Special Right Triangles 30°-60°-90° right triangle In a 30°-60°-90° right triangle, the length of

Special Right Triangles 30°-60°-90° right triangle 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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Converse of the Pythagorean Theorem If ABC is a triangle with sides of lengths

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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 14 -10 Determine if the following can be the lengths of the sides

Example 14 -10 Determine if the following can be the lengths of the sides of a right triangle: a. 51, 68, 85 b. 2, 3, yes Copyright © 2013, 2010, and 2007, Pearson Education, Inc. c. 3, 4, 7 no

The Distance Formula: An Application of the Pythagorean Theorem Copyright © 2013, 2010, and

The Distance Formula: An Application of the Pythagorean Theorem Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

The Distance Formula The distance between the points A(x 1, y 1) and B(x

The Distance Formula The distance between the points A(x 1, y 1) and B(x 2, y 2) is given by Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 14 -11 Show that A(7, 4), B(– 2, 1), and C(10, − 4)

Example 14 -11 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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 14 -11 (continued) ABC is a right triangle with hypotenuse BC. Copyright ©

Example 14 -11 (continued) ABC is a right triangle with hypotenuse BC. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 14 -12 Determine whether the points A(0, 5), B(1, 2), and C(2, −

Example 14 -12 Determine whether the points A(0, 5), B(1, 2), and C(2, − 1) are collinear. 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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 14 -12 (continued) Since AB + BC = AC, the points are collinear.

Example 14 -12 (continued) Since AB + BC = AC, the points are collinear. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Using the Distance Formula to Develop the Equation of a Circle From the distance

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 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Using the Distance Formula to Develop the Equation of a Circle From the distance

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 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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Chapter 14 Area Pythagorean Theorem and Volume CopyrightChapter 14 Area Pythagorean Theorem and Volume Copyright