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TEN FOR TEN® DRAW IT! 1) 2) 3) In the standard (x, y) coordinate plane, 3 corners of a rectangle are (1, -1), (-4, -1), and (1, 4). Where is the rectangle’s fourth corner? a) (1, 4) c) (-1, 1) b) (-1, 4) d) (-1, -4) e) (-4, 4) A certain circle has an area of 25π square inches. How many inches long is its circumference? a) 5π c) 15π b) 10π d) 20π e) 25π Points A (-3, -4) and B (7, -2) determine line segment AB in the standard (x, y) coordinate plane. If the midpoint of AB is (a, -3), what is the value of a? a) 2 c) 4 b) -4 d) -5 e) 5 PLEASE CHECK THE ANSWERS AND EXPLANATIONS FOR PROBLEMS 1-3 NOW 4) 5) 6) 7) An object detected on radar is 5 miles to the east, 4 miles to the north, and 1 mile above the tracking station. Among the following, which is the closest approximation to the distance, in miles, that the object is from the tracking station? a) 6 c) √47 b) √42 d) 9 e) √92 In the standard (x,y) plane, the triangle with vertices at (0,0), (0,k), and (2,m), where m is a constant, changes shape as k changes. What happens to the triangle’s area, expressed in square coordinate units, as k decreases starting from 2? a) The area decreases as k increases c) The area always equals 2. b) The area decreases as k decreases d) The area always equals m e) The area decreases and then increases as k decreases If the length of a square is increased by 2 inches and the width is increased by 3 inches, a rectangle is formed. If each side of the original square is x inches long, what is the area of the new rectangle, in square inches? a) x + 5 c) x2 + 6 b) 2x + 6 d) x2 + 5x + 6 e) x2 + 6x + 5 The length of a rectangle is (x + 5) units and its width is (x + 9) units. Which of the following expresses the remaining area of the rectangle, in square units, if a square (x – 1) units in length is removed from the interior of the rectangle? a) 13 c) 2x + 13 b) 44 d) 12x + 44 e) 16x + 44 DRAW IT! 2 8) 9) 10) The lengths of the sides of a triangle are 3, 8, and 9 inches. How many inches long is the longest side of a similar triangle that has a perimeter of 60 inches? a) 9 c) 20 b) 11 d) 24 e) 27 The measure of the vertex angle of an isosceles triangle is (x – 20)°. The base angles each measure (2x + 30)°. What is the measure in degrees of one of the base angles? a) 8° c) 42.5° b) 28° d) 47.5° e) 86° A circle with center (-3,4) is tangent to the x-axis in the standard (x,y) coordinate plane. What is the radius of this circle? 10/30/09 a) 3 c) 5 b) 4 d) 9 e) 16 TEN FOR TEN® ANSWERS AND EXPLANATIONS DRAW IT! E. Once you’ve drawn the grid and put in the given three points, it’s pretty simple to see where the fourth one needs to be. If you tried to get this right (and didn’t) without drawing, make “drawing it” part of your technique for solving these problems, because rushing will never make you faster. (Note that the upper left-hand point is red!) 1) A = 25 pi r = 5, so d = 10; which means C = 10 pi 2) (7,-2) 3) (-3,-4) (a,-3) B. SAT circle problems require you to convert the information you’re given into the information you need using formulas that you may know well (and if you don’t, they’re printed out for you in the “formula strip” at the beginning of every SAT math section). While you’re practicing with TEN FOR TENs, consult the Math Companion whenever you’re unsure—using a formula again and again is the best way to learn it. A. Here, note that the y-value of the unknown point is -3, which is halfway between the y-values of the known points. Since that’s the case, won’t the x-value of the unknown point also be half way between the known x-values? If logic has anything to do with math, it will. PLEASE RETURN TO WORK ON PROBLEMS 4-10 NOW DRAW IT! ANSWERS AND EXPLANATIONS 2 1 root 42 root 41 5 4) 0,k 0,0 5) 10/30/09 2,m 4 B. In order to solve this one, we need to draw a three-dimensional figure in a two-dimensional space. There are many ways to do so, but let me explain the drawing to the left. There is a right angle between the “5” and “4” sides (imagine “east” and “north”); also, there is a right angle between the “4” and “1” side (which is “up”). So, to find the distance (purple dashed line) between the object and the station, we first have to find out how far a spot on the ground directly under the object would be from the station. We can do so by using the Pythagorean Theorem (52 + 42 = c2), making that distance √41; now, let’s use the Theorem again with sides √41 and 1; the result is √42. E. Once you’ve put (0,0) and (0,k) into the grid (remember, the question asks what happens as k decreases starting from 2), you realize that the third point (2,m) can go anywhere along the line x = 2. I have chosen (2, 0)—but this will work just as well if you chose another y-value for m. Note that as k decreases, the triangle becomes smaller at first (as k approaches zero) and then bigger (as k moves away from zero down the y-axis). If you chose (b), you probably didn’t draw it. If you chose (a), you missed “k decreases” in the problem. DRAW IT! ANSWERS AND EXPLANATIONS 3 (x + 3) 6) (x + 2) D. You probably could have solved this without drawing it. But drawing puts all the information you need right in front of you, making it much less likely that you’ll make a “dumb” mistake. x x (x + 9) (x - 1) (x + 5) 7) (x - 1) 8 3 8) 9 10/30/09 E. The big rectangle measures x2 + 14x + 45, right? The small square measures x2 - 2x + 1. So, subtracting the second from the first gives us ... [what happens when we subtract a negative number?]. ALTERNATIVELY, how about we sub in 3 for x, which makes the big rectangle 12 by 8, or 96 square inches, and the small square, which we must subtract, 2 by 2, or 4 square inches. So, the rectangle minus the square is 92 square inches. Sub in 3 for x in the answer choices. E. Drawing this triangle can lead to the next logical step, which is adding the sides to get a perimeter of 20. How does this relate to a similar triangle with a perimeter of 60? Wouldn’t we just need to multiply each side by 3? Yep. DRAW IT! ANSWERS AND EXPLANATIONS 4 (x - 20o) 9) E. Again, you could have imagined the whole thing, added up the x’s and numbers, come up with 5x + 40 = 180; or x = 28. Then, without any visual aid, you could have multiplied x by 2 and added 30 to get the answer. Is that the way it worked out? (2x + 30o) B. People who get problems like this wrong “imagine” which coordinate measures the distance from, in this case, the x-axis. Why imagine when you can draw? 10) 4 10/30/09