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

Chapter

Area, Pythagorean Theorem, and

Volume

1414

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

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


Parts of a Right Triangle


Pythagorean Theorem

Given a right triangle with legs a and b and hypotenuse c, c2 = a2 + b2.

If BC = 3 cm and AC = 4 cm,

what is the length of AB?

5 cm


Example 14-7b

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. 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?


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.


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.


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


Converse of the Pythagorean Theorem

If ABC is a triangle with sides of lengths a, b, and c such that c2 = a2 + b2, then ABC is a right triangle with the right angle opposite the side of length c.


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, c. 3, 4, 7

yes yes no


The Distance Formula: An Application of the Pythagorean Theorem


The Distance Formula

The distance between the points A(x1, y1) and B(x2, y2) is given by


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 14-11


Example 14-11 (continued)

ABC is a right triangle with hypotenuse BC.


Determine whether the points A(0, 5), B(1, 2), and C(2, −1) are collinear.

Example 14-12

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.


Example 14-12 (continued)

Since AB + BC = AC, the points are collinear.


Using the Distance Formula to Developthe 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


Using the Distance Formula to Developthe Equation of a Circle

From the distance formula, we

have

The equation of a circle with the center (h, k) and

radius r is


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

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