GRE TEST prep Geometry Beth Bowman, Ph.D. VSSA GRE
Description: GRE TEST prep Geometry Beth Bowman, Ph.D. VSSA GRE prep course Areas of Knowledge Lines Lines 180 on straight line K 242 2: In the diagram, L1 is parallel to L2. The measure of angle q is 40. What is the sum of the measures of the acute
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slide1. GRE TEST prep Geometry Beth Bowman, Ph.D.
VSSA GRE prep course<br>
slide2. Areas of Knowledge<br>
slide3. Lines Lines
180° on straight line
K 242 2: In the diagram, L1 is parallel to L2. The measure of angle q is 40°. What is the sum of the measures of the acute angles shown in the diagram?
K 263 22: Which of the following statements must be true about the figure shown to the right if L1 is parallel to L2?
x = a
x = b
a = b
y = b
x + y = a + b a=b
c=d a=b
c=d If two lines parallel a + c = 180
a + d =180<br>
slide4. Triangles =180°
Equilateral
Isosceles
Right (next page)
Area = 1/2bh
Angle/side relationship
Longest side opposite longest angle, etc…
Length of longest side < length of other sides sum
PR 272: A triangle has sides 4, 7, and x. Which of the following could be the perimeter of the triangle?
13
16
17
20
22
PR 296 10: Compare
x
5.9 3 9 x<br>
slide5. Right Triangles Pythagorean theorem a2 + b2 = c2
3, 4, 5
5, 12, 13
PR 276: What is the area of the rectangle above?
PR 297 13: In the rectangle ABCD above, what is the area of the triangle ABD?
Right Isosceles
x, x, x√2
30:60:90
x, x√3, 2x
PR 279: Triangle XYZ is an equilateral triangle. If the perimeter of the triangle is 12, what is its area?
2√3
4√3
8
12
8√3 6 10 x x x√2 x 2x x√3 30° 60° 45° 45° 3 5 B C D A<br>
slide6. Triangle Qs K 199
In the triangle above, what is the length of side BC?
K 204 1: In the triangle ABC, if AB=4, then AC=
6
7
8
9
10
K 296 36: In the figure below, what is the value of x? 70° 30° x° C<br>
slide7. Circles Circle
Circumference (C) = 2πr=πd
Arc length = C (x°/360)
PR 297 15: The circumference of the circle with center O shown above is 15π. LMNO is a parallelogram and angle OLM = 108°. What is the length of the minor arc?
15π
9π
3π
2π
π
Area = πr2
Area of sector = area * (x°/360)
PR 296 8: Compare
The circumference of a circular region with radius r
The perimeter of a square with side r
Concentric, circumscribed, inscribed
If inscribed triangle and 1 side is diameter, it is a right triangle x° c O B A N M L<br>
slide8. Circles Qs K 226: The diameter of circle O is D, and the area is A
Compare
(πD2)/2
A
K 261 10. A and B are points on the circumference of the circle with center O. The length of chord AB is 15.
Compare
Circumference of circle O
12π
K 264 28: The circle shown has center T. The measure of angle TVU is 60°. If the circle has a radius of 3, what is the length of segment RS?
2
2√2
3
3√3
6√2 T R U S V 60°<br>
slide9. Other shape things Total angles in shape = (#sides -2) * 180
Square: area and perimeter
Trapezoid: area (1/2)(b1 + b2)(h) and perimeter
3D objects
Volume: area of object x height
Surface area: Sum of all areas of sides
ETS 149 12. In the rectangular solid above, TU = 3, UV = 4, and VR = 2. What is the area of the shaded rectangular region?
What is the surface area and volume of this object? b1 b2 h c c 5 3 V R S T U<br>
slide10. Overview questions PR 283: In the figure above, a circle with the center O is inscribed in square WXYZ. If the circle has radius 3, then PZ =
6
3√2
6 + √2
3 + √3
3√2 + 3
K 207: In the figure below, if the area of the circle with center O is 9π, what is the area of triangle POQ?
4.5
6
9
3.5π
4.5π W X Y Z P O Q P O<br>
slide11. Overview questions PR 289: What is the length of the longest distance between any two corners in a rectangular box with dimensions 3 inches by 4 inches by 5 inches?
PR 291: Compare
The diameter of a circle with area 49π
14
PR 292: D is the midpoint of AC
Compare
m
n m C A B 60° n D<br>
slide12. Overview questions K 291 11: AD = DC
Compare
The area of ΔBAD
The area of ΔBCD
K 293 19: The perimeter of a square equals the perimeter of a rectangle that is not a square.
Compare
The area of the square
The area of the rectangle
K 294 27: A square has the same area as a rectangle whose length is 2 more than its width. If the perimeter of the rectangle is 12, what is the perimeter of the square?
6
6√2
8
8√2
10 C A B D<br>
slide13. Overview questions K 212 1: What is the value of x in the figure above?
4
3√3
3√5
5√3
9
K 230 5
Compare
Perimeter of square A / Perimeter of square B
Length of WY / Length of PR
K 290 5
Compare
z
20
ETS 155 1: In the figure, ABCD is a parallelogram
Compare
The area of ABCD
24 F E 5 x 8 H G 6 W X Z Y Square A P Q S R Square B 4z° 5z° 125° 6 4 A D B C<br>
slide14. PR 295 6: The circle with center D is drawn inside the circle with center C, as shown in the figure. If CD = 3, which is the area of semicircle EAB?
(9/2) π
9π
12π
18π
36π
K 261 12: ΔABC lies in the xy-plane with C at (0,0), B at (6, 0), and A at (x, y), where x and y are positive. The area of ΔABC is 18 square units.
Compare
y
6 Overview questions<br>
slide15. Overview questions PR 316 1: Line AB is tangent to the circle C at point A. The radius of the circle with center C is 5 and BC = (10√3)/3
Compare
The length of line segment AB
The length of line segment AC
PR 318 9: What is the area of triangle ABC?
2
4
4√2
7
8 A C B 4 3 4<br>
VSSA GRE prep course<br>
slide2. Areas of Knowledge<br>
slide3. Lines Lines
180° on straight line
K 242 2: In the diagram, L1 is parallel to L2. The measure of angle q is 40°. What is the sum of the measures of the acute angles shown in the diagram?
K 263 22: Which of the following statements must be true about the figure shown to the right if L1 is parallel to L2?
x = a
x = b
a = b
y = b
x + y = a + b a=b
c=d a=b
c=d If two lines parallel a + c = 180
a + d =180<br>
slide4. Triangles =180°
Equilateral
Isosceles
Right (next page)
Area = 1/2bh
Angle/side relationship
Longest side opposite longest angle, etc…
Length of longest side < length of other sides sum
PR 272: A triangle has sides 4, 7, and x. Which of the following could be the perimeter of the triangle?
13
16
17
20
22
PR 296 10: Compare
x
5.9 3 9 x<br>
slide5. Right Triangles Pythagorean theorem a2 + b2 = c2
3, 4, 5
5, 12, 13
PR 276: What is the area of the rectangle above?
PR 297 13: In the rectangle ABCD above, what is the area of the triangle ABD?
Right Isosceles
x, x, x√2
30:60:90
x, x√3, 2x
PR 279: Triangle XYZ is an equilateral triangle. If the perimeter of the triangle is 12, what is its area?
2√3
4√3
8
12
8√3 6 10 x x x√2 x 2x x√3 30° 60° 45° 45° 3 5 B C D A<br>
slide6. Triangle Qs K 199
In the triangle above, what is the length of side BC?
K 204 1: In the triangle ABC, if AB=4, then AC=
6
7
8
9
10
K 296 36: In the figure below, what is the value of x? 70° 30° x° C<br>
slide7. Circles Circle
Circumference (C) = 2πr=πd
Arc length = C (x°/360)
PR 297 15: The circumference of the circle with center O shown above is 15π. LMNO is a parallelogram and angle OLM = 108°. What is the length of the minor arc?
15π
9π
3π
2π
π
Area = πr2
Area of sector = area * (x°/360)
PR 296 8: Compare
The circumference of a circular region with radius r
The perimeter of a square with side r
Concentric, circumscribed, inscribed
If inscribed triangle and 1 side is diameter, it is a right triangle x° c O B A N M L<br>
slide8. Circles Qs K 226: The diameter of circle O is D, and the area is A
Compare
(πD2)/2
A
K 261 10. A and B are points on the circumference of the circle with center O. The length of chord AB is 15.
Compare
Circumference of circle O
12π
K 264 28: The circle shown has center T. The measure of angle TVU is 60°. If the circle has a radius of 3, what is the length of segment RS?
2
2√2
3
3√3
6√2 T R U S V 60°<br>
slide9. Other shape things Total angles in shape = (#sides -2) * 180
Square: area and perimeter
Trapezoid: area (1/2)(b1 + b2)(h) and perimeter
3D objects
Volume: area of object x height
Surface area: Sum of all areas of sides
ETS 149 12. In the rectangular solid above, TU = 3, UV = 4, and VR = 2. What is the area of the shaded rectangular region?
What is the surface area and volume of this object? b1 b2 h c c 5 3 V R S T U<br>
slide10. Overview questions PR 283: In the figure above, a circle with the center O is inscribed in square WXYZ. If the circle has radius 3, then PZ =
6
3√2
6 + √2
3 + √3
3√2 + 3
K 207: In the figure below, if the area of the circle with center O is 9π, what is the area of triangle POQ?
4.5
6
9
3.5π
4.5π W X Y Z P O Q P O<br>
slide11. Overview questions PR 289: What is the length of the longest distance between any two corners in a rectangular box with dimensions 3 inches by 4 inches by 5 inches?
PR 291: Compare
The diameter of a circle with area 49π
14
PR 292: D is the midpoint of AC
Compare
m
n m C A B 60° n D<br>
slide12. Overview questions K 291 11: AD = DC
Compare
The area of ΔBAD
The area of ΔBCD
K 293 19: The perimeter of a square equals the perimeter of a rectangle that is not a square.
Compare
The area of the square
The area of the rectangle
K 294 27: A square has the same area as a rectangle whose length is 2 more than its width. If the perimeter of the rectangle is 12, what is the perimeter of the square?
6
6√2
8
8√2
10 C A B D<br>
slide13. Overview questions K 212 1: What is the value of x in the figure above?
4
3√3
3√5
5√3
9
K 230 5
Compare
Perimeter of square A / Perimeter of square B
Length of WY / Length of PR
K 290 5
Compare
z
20
ETS 155 1: In the figure, ABCD is a parallelogram
Compare
The area of ABCD
24 F E 5 x 8 H G 6 W X Z Y Square A P Q S R Square B 4z° 5z° 125° 6 4 A D B C<br>
slide14. PR 295 6: The circle with center D is drawn inside the circle with center C, as shown in the figure. If CD = 3, which is the area of semicircle EAB?
(9/2) π
9π
12π
18π
36π
K 261 12: ΔABC lies in the xy-plane with C at (0,0), B at (6, 0), and A at (x, y), where x and y are positive. The area of ΔABC is 18 square units.
Compare
y
6 Overview questions<br>
slide15. Overview questions PR 316 1: Line AB is tangent to the circle C at point A. The radius of the circle with center C is 5 and BC = (10√3)/3
Compare
The length of line segment AB
The length of line segment AC
PR 318 9: What is the area of triangle ABC?
2
4
4√2
7
8 A C B 4 3 4<br>