1. 14 Chapter Area, Pythagorean Theorem, and Volume
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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
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3. Parts of a Right Triangle
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4. 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
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5. 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.
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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?
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7. 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 feet tall.
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8. Special Right Triangles
The length of the hypotenuse in a 45°-45°-90° (isosceles) right triangle is times the length of a leg.
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9. 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.
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10. 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.
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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
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12. The Distance Formula: An Application of the Pythagorean Theorem
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The Distance Formula
The distance between the points A(x1, y1) and B(x2, y2) is given by
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14. 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.
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Example (continued)
ABC is a right triangle with hypotenuse BC.
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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.
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17. Example (continued)
Since AB + BC = AC, the points are collinear.
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18. 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
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19. 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
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