Geometry: Points and Space (xy, Locus, Spatial Ability)
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1. The base of an isosceles triangle has endpoints at coordinates (1,1) and (13,1). The area of the triangle is 48. The third point of the triangle could be located at:

(A) (6, –6)
(B) (6, –11)
(C) (7, 9)
(D) (7, –11)
(E) (13, 13)

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The base is 12 units long, half the base is 6 units, so the height has to be 8 units. To get a height of 8, the y-coordinate of the third point is either 9 or –7. So the answer is (C). Note that since 7 is halfway between 1 and 13, the triangle is indeed isoceles.

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2. What is the area of the triangle formed by the x-axis, the line y = x, and the line y = –(1/2)x + 3?

(A) 3
(B) 6
(C) 9
(D) 12
(E) 15

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It's best to draw this one. The line with the negative slope hits the y-axis at (0,3) and hits the x-axis at (6,0). You have to find where the lines y=x and y=–(1/2)x+3 meet. Plug the first equation into the second to get x=–(1/2)x+3. You get x = 2. So the lines meet at (2,2). The triangle is made up of the points (0,0), (2,2) and (6,0). The base is 6, the height is 2 and the area is 6. The answer is (B).

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3. A rectangle has a perimeter of 20 inches. Which of the following could be the area of this rectangle in square inches?

I. 8
II. 23
III. 26

(A) I, only
(B) II, only
(C) I and II, only
(D) II and III only
(E) I, II, and III

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If the rectangle is a 5 by 5 square, the area is 25 and this is the largest possible area. Any smaller area is possible (a long, skinny rectangle would have an area arbitrarily close to zero). So 8 and 23 are possible areas and the answer is (C).

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4. Points A and B on plane P are 5 units apart. How many points in plane P are both 3 units from point B and 6 units from point A?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

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Draw A and B 5 units apart. Draw a circle with radius 3 around B and another circle with radius 6 around A. The circles intersect in two places so the answer is (C).

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5. Points A and B on plane P are 5 inches apart. How many points in plane P are 4 inches from point B and more than 4 inches from point A?

(A) 0
(B) 1
(C) 2
(D) 3
(E) more than 3

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Draw the two circles of radius 4, one around B and the other around A. How many points are on the circle around B and outside the circle around A? There's a whole section of the circle around B that fits this description and this section contains an infinite number of points. So the answer is (E).

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6. Points A and B are 6 inches apart. Point M is the midpoint of AB. Point C is 4 inches from point M and point D is 2 inches from point B. Points A, B, C, D, and M all lie on a plane. Which of the following could be the length of CD in inches?

I. 0
II. 1
III. 9

(A) none
(B) I, only
(C) II, only
(D) I and II, only
(E) I, II, and III

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Draw line AB from left to right. Put M in the middle and label AM and BM with their lengths (3). Then draw a big circle around M with a radius of 4 (the circle contains the line AB has 1 inch to spare around points A and B). Now draw a circle around B with a radius of 2 (it crosses the big circle at two points). Since C and D could be the same point, the length of CD could be 0. If D is as far to the right as it can be and C is as far to the left as it can be, they will be 9 units apart. The length of CD can be anything between 0 and 9 inclusive. So the answer is (E).

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7. Points A, B, C, and D lie on a plane. The distance between points A and B is 24, AC = BC = 15, and AD = BD = 13. Which of the following could be the length of CD?

I. 4
II. 9
III. 14

(A) II, only
(B) III, only
(C) II and III, only
(D) I and III, only
(E) I and II, only

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Draw line AB and label it with its length (24). Then draw point C above the midpoint of AB and label AC and BC with their lengths (15). The right triangle formed by points A, C, and the midpoint of AB is a 9-12-15 triangle (a variant of the 3-4-5). So distance from point C to the midpoint of AB is 9. Point D is going to be 5 units above or below the line because it will make a 5-12-13 triangle with the midpoint of AB. If point D is 5 units above then CD = 4. If point D is 5 units below then CD = 14. So I and III are the only possibilities and the answer is (D).

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8. In a diagram (not shown) points A, B, C, D, E, and F are endpoints of five line segments that lie in a plane and intersect at point A. Line segment AD bisects angle BAC, line segment AE bisects angle DAC, and line segment AF bisects angle EAC. Angle BAC is less than 90 degrees. If all angles less than 90 degress and greater than zero degrees in the diagram are measured, including overlapping and non-overlapping angles, how many numerically different results will be obtained?

(A) 10
(B) 9
(C) 8
(D) 7
(E) 6

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This is a "fan" with point A as the vertex and B, D, E, F, and C forming the outside of the fan. (This is a "draw it or die" question.) Suppose BAC (the largest angle) measures 1 unit. The smallest angle you can make (FAC or EAF) is 1/8. You can also make 2/8, 3/8, 4/8, 6/8, and 7/8 for a total of 7 different angles (including BAC). The answer is (D).

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9. Each section of one side of a square card is painted a different color. Which of the choices below could be the result of rotating the card represented in the figure above by 270° counter-clockwise about an axis perpendicular to the plane of the card that passes through its center?

(A)
(B)
(C)
(D)
(E)

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Choices D and E could not be produced by rotating the card (if you go clockwise from red it always has to be red-pink-gray-blue no matter how you rotate it). Choice C is produced by a 90 degrees counterclockwise rotation. Choice B is a 180 degree rotation. The answer is (A).

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10. A cube has a different letter on each face. The letters are A, B, C, D, E, and F. Another cube is similar except that the letters on the faces are A, B, C, G, H, and I. Both cubes are opaque. The cubes are glued together face-to-face and set down on an opaque table. If the cubes are not moved, what is the minimum number of different letters that must be visible to a person who walks around the table? (The person can see all of the exposed faces.)

(A) 5
(B) 6
(C) 7
(D) 8
(E) 9

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To get the minimum number of different letters, you want to cover up as many letters as possible. Stay away from A, B, and C because if you cover up an A and a B for example you still have an A and a B visible. If you glue a D face to a G face and place the blocks so that the E and H faces are on the opaque table, then only A, B, C, F, and I will be visible. That's the best you can do. The answer is (A).

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