Rectangles and Squares
I. Line I Rectangle
III. Circle
(A) I only
2 distinct points, A and B, are at equal distance from a third point C. On which of the following shapes can A, B, and C be?
I. Line II. Rectangle III. Circle
(A) I only
What is the ratio of the area of the smallest shaded circle to the area of the shaded crescent?
A B
D C
O
Figure is not drawn to scale.
35o
x
E F
In the figure, ABCD is a rectangle and AE = EO = FO = FB.
Figure is not drawn to scale.
ABFG and BCDE are 2 squares with areas a and 2a, and the area of the rectangle ABCD is 10.00. What is the parallel tangent lines to 3 circles, as shown in the figure. They are separated from each other by an equal distance, s.
6. (Hard)
7. (Hard)
Provide your answer in terms of w and p.
Trigonometry
1. (Medium)
2. (Medium)
3. (Hard)
4. (Hard)
5. (Hard)
Coordinate Geometry
1. (Medium)
Figure not drawn to scale.
A B
C
3 is an equilateral
triangle with side length . As shown in the figure, a circle is inscribed inside
.
What is the area of the circle?
∆ABC 3
∆ABC
.
wA small wheel of radius w is rolling on a circular path of radius p.
What percentage of the circumference
of the circle O is traveled by the wheel,
.
Op
2
A 4 B
D C
60o 3
Figure is not
What is the area of the trapezoid ABCD?
drawn to scale.
a c b
Figure is not
In the figure, if c/b = 1/2, what is a/c?
drawn to scale.
2
Figure is not drawn to scale.
In the figure, AD = ?
30o
A B
C
D
A B
D C
E
F
60o 8
Figure is not drawn to scale.
ABCD is a rectangle. If DE = EA, AB = ?
A
B C D
60o F E
5
Figure is not drawn to scale
In the figure, ABCD is a rectangle.
EF = ?
A B
O
C(3, r) D
In the figure, ABCD is a square and OA = OB.
What is the value of r?
y
x
Private Tutor for SAT Math Success 2006 | Geometry 6 - 51 2. (Medium)
3. (Hard)
What is the area of the circle O?
Symmetry
1. (Easy)
How many lines of symmetry an isosceles triangle has?
2. (Easy)
A and B are two points on a number line with coordinates -5 and 27, respectively. What is the coordinate of the point around which A and B are symmetrical?
3. (Easy)
4. (Medium)
On the xy-plane, the coordinates of point A is (-3, 1).
What is the addition of the x- and y-coordinates of the point B that is symmetric to A around point C(1, -1)?
5. (Medium)
On the xy-plane, the coordinates of point A is (-3, 1).
What is the multiplication of the x- and y-coordinates of the point B that is symmetric to A around the y = x line?
6. (Medium)
O B(-3,1)
C(-2, r)
In the figure, BC= OB.
What is the value of r?
Figure is not drawn to scale.
x y
.
A
B
is an equilateral C
triangle. and are tangents to the circle O. The coordinates of the centre of the circle and the point A are:
(3, -2) and (0, -2), respectively.
∆ABC
AC AB x
-y O
W
Y X Z
Which of the following pairs are not symmetric around any axis?
(A) W, Z (B) Z, Y (C) Y, X (D) X, W (E) W, Y
x y
The triangle in the figure is equilateral. Which of the following will be obtained if the object is rotated 30o counter clockwise, around an axis perpendicular to the plane of the triangle and passing through the point of intersection of the 3 line segments in the figure?
(A)
(B)
(C)
(D)
(E)
3 - Dimensional Objects
1. (Easy)
For a cube of edge length a, what is the total length of all the edges?
2. (Medium)
3. (Medium)
a. The cut through the ABCD plane.
b. The cut through the EFGH plane.
4. (Medium)
5. (Medium)
A cylinder is inscribed inside a cube of edge length 5.
What is the maximum value that the volume of the cylinder can take?
6. (Hard)
What is the length of the longest line segment you can fit inside of a cube with side length a?
7. (Hard)
8. (Hard)
Create your own questions and answer them by replacing the cubes in questions 1, 2, 3, 5, 6 and 7, with a rectangular prism with side lengths a, b and c.
Note that cube is a special rectangular prism with a = b = c.
A
B C
D a
Figure is not drawn to scale.
Consider the cube in the figure. What is the area of ABCD?
A
B C
D a
E
The cube in the figure is cut in two different ways to create two objects in each case. Which cut creates the maximum number of
corners? H
G F
h h
h
Figure is not drawn to scale.
A cylinder is cut though the dotted line shown in the figure, to obtain two cylinders.
What is the ratio of the total surface area of the final two cylinders to the surface area of the original cylinder?
Hint:
Base diam ete r and
the height of the cylinder are both 5.
A
B C
D a
If the cube in the figure is cut through the plane ABCD, what would be the difference between the original surface area of the cube and the total surface areas of the two resultant triangular prisms combined?
Hint : There is
no need to calculate the surface of the original cube are also the areas of all 3 solids before and after the cut. Since all the surfaces surfaces of the two tria
ngular prisms, the only extra surface to consider is the surface
of the cut.
Private Tutor for SAT Math Success 2006 | Geometry 6 - 53 9. (Hard)
10. (Hard)
What is its surface area of the pyramid in the previous question?
11. (Hard)
You are making a cradle-shape paper object with length l, width w and height h . If l > w and the height remains the same for the body and the ends of the cradle, which of the following will be the most suitable cut?
(E) None of the above.
a h
A
C
O a
The object shown in the figure is E
called a square right pyramid. It has a square base (ABCD) of edge length a and height OE = h. is perpendicular to the base of the pyramid.
CE = ?
OE
B D
(A)
(B)
(C)
(D)