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Chapter 13

# Which point lies in the yz plane and xz planes draw a

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Unformatted text preview: that the triangle with vertices P 2, 4, 0 , Q 1, 2, 1, 1, 2 is an equilateral triangle. 1, 8. Find the lengths of the sides of the triangle with vertices A 1, 2, 3 , B 3, 4, 2 , and C 3, 2, 1 . Is ABC a right triangle? Is it an isosceles triangle? 9. Determine whether the points lie on a straight line. (a) A 5, 1, 3 , B 7, 9, 1 , C 1, 15, 11 (b) K 0, 3, 4 , L 1, 2, 2 , M 3, 0, 1 10. Find the distance from 3, 7, (a) The xy-plane (c) The xz-plane (e) The y-axis 5 (b) (d) (f ) to each of the following. The yz-plane The x-axis The z-axis 11. Find an equation of the sphere with center 1, 4, 3 and radius 5. What is the intersection of this sphere with the xz-plane? 12. Find an equation of the sphere with center 6, 5, 2 and radius s7. Describe its intersection with each of the coordinate planes. x 4z ■ z y 2 8x ■ 11 16y ■ ■ 1 ■ ■ ■ ■ ■ ■ P1 x 1, y1, z1 to P2 x 2 , y2 , z2 is x1 x2 2 y1 , 2 y2 z1 , z2 2 (b) Find the lengths of the medians of the triangle with vertices A 1, 2, 3 , B 2, 0, 5 , and C 4, 1, 5 . represented by the equa- 4 represent in 2 ? What does 3 it represent in ? Illustrate with sketches. (b) What does the equation y 3 represent in 3 ? What does z 5 represent? What does the pair of equations y 3, z 5 represent? In other words, describe the set of points x, y, z such that y 3 and z 5. Illustrate with a sketch. 4y 2 2z 19. (a) Prove that the midpoint of the line segment from 2. 6. (a) What does the equation x 2 ■ and xz-planes? Draw a rectangular box with the origin and 2, 3, 5 as opposite vertices and with its faces parallel to the coordinate planes. Label all vertices of the box. Find the length of the diagonal of the box. and R 833 Exercises 1. Suppose you start at the origin, move along the x-axis a tion x ❙❙❙❙ 20. Find an equation of a sphere if one of its diameters has end- points 2, 1, 4 and 4, 3, 10 . 21. Find equations of the spheres with center 2, 3, 6 that touch (a) the xy-plane, (b) the yz-plane, (c) the xz-plane. 22. Find an equation of the largest sphere with center (5, 4, 9) th...
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