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TEN FOR TEN®
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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
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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
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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
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ANSWERS AND EXPLANATIONS
2
1
root 42
root 41
5
4)
0,k
0,0
5)
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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.
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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
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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.
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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
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