By ed van der Geer at al Birkhaeuser

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**Additional resources for Arithmetic Algebraic Geometry**

**Sample text**

Now Work 17 PROBLEMS AND 27 1 Use the Distance Formula If the same units of measurement, such as inches, centimeters, and so on, are used for both the x-axis and y-axis, then all distances in the xy-plane can be measured using this unit of measurement. E XA MP LE 2 Finding the Distance between Two Points Find the distance d between the points 11, 32 and 15, 62. Solution First plot the points 11, 32 and 15, 62 and connect them with a straight line. See Figure 9(a). To find the length d, begin by drawing a horizontal line from 11, 32 to 15, 32 and a vertical line from 15, 32 to 15, 62, forming a right triangle, as in Figure 9(b).

A5–A6) 6. True or False Two triangles are congruent if two angles and the included side of one equals two angles and the included side of the other. (pp. A16–A17) 3. If 3 and 4 are the legs of a right triangle, the hypotenuse is _________. (pp. A13–A14) 4. Use the converse of the Pythagorean Theorem to show that a triangle whose sides are of lengths 11, 60, and 61 is a right triangle. (p. A14) 14 CHAPTER 1 Graphs Concepts and Vocabulary 7. If 1x, y2 are the coordinates of a point P in the xy-plane, then x is called the of P and y is the _____________ of P.

Solution Figure 32 Replace y by -y. Since -y = x3 is not equivalent to y = x3, the graph is not symmetric with respect to the x-axis. y-Axis: Replace x by -x. Since y = ( -x)3 = -x3 is not equivalent to y = x3, the graph is not symmetric with respect to the y-axis. Origin: Replace x by -x and y by -y. Since -y = ( -x)3 = -x3 is equivalent to y = x3 (multiply both sides by -1), the graph is symmetric with respect to the origin. x-Axis: y 8 (0, 0) Ϫ6 (Ϫ1, Ϫ1) (Ϫ2, Ϫ8) Ϫ8 First, find the intercepts of y = x3 .