GRE Geometry

Question 1

What is the sum of the angles of a triangle?

Answer 1

180

Question 2

2 angles in every triangle have to be acute: T of F

Answer 2

True

Question 3

What can you say about the 3 sides of a triangle?

Answer 3

• the sum of 2 sides of triangle must be greater than the 3rd side
• ## P - Q < X < P + Q (X is the third side and must satisfy these values)

Question 4

What can you say about an isosceles triangle?

Answer 4

• sides are equal and 2 angles are equal
• 1 - 1 - radical 2 triangle or has the same proportions
• 45-45-90 it could be sometimes, but not always

Question 5

Area of a triangle

Answer 5

A = 1/2bh

• base can be any side
• h is called the altitude
• you can also extend a triangle to find these values

Question 6

Pythagorean theorem

Answer 6

a2 = b2 + c2 
a = hypotenuse 
b, c are two sides 
- this only works for a right triangle 
- this won't work for other types of triangles

Question 7

What are common pythagorean triplets?

Answer 7

3-4-5
5-12-13
8-15-17
7-24-25
9-40-41
11-60-61
12-35-37
13-84-85

If you scale any of these by a scale factor (multiply all sides by the same value.

Question 8

How can you use proportional reasoning in triangles?

Answer 8

When solving for one side in the Pythagorean theorem, you can scales the sides down by dividing the values to reduce them (using the GCF), in order to make the values smaller to work with. Solve for the third side and then scale it up again via multiplication using the same value you scaled down by.

Question 9

What can you assume about similar triangles?

Answer 9

• they are the same geometric shape, but with different scales for the sides
• they have equal angles for all the three angles
• sides are proportional and contain a scale factor
• Scale factor is found through the proportion:
  Bigger triangle side / smaller triangle side = scale factor
• If just two pair angles equal each other across two triangles, then they are similar triangles
• if there is a smaller triangle within a larger triangle and set of the sides are parallel, then the smaller within could be a similar triangle to the larger

Question 10

What can you say about the scale factor for Areas of triangles?

Answer 10

Given the scale factor of between 2 shapes, then we can determine the scale factor in areas:
- A = 1/2bh
both h and b (altitude and a side) can be multiplied by the scale factor k

• k^2 is the scale factor for areas of two triangles. Find the scale factor across the two triangles, square it, and then multiply the smaller area by it to get the larger area (divide it into the larger area to get the smaller area)

Question 11

What can we say about equilateral triangles?

Answer 11

• all 3 sides are equal
• all 3 sides have equal angles and must be 60 degrees each
• cut it in half to get a 30-60-90 triangle
• A = s^2radical 3 / 4
• every special fact about iscoceles triangles also applies to equalateral triangles

Question 12

What are the 4 lines of a triangle?

Answer 12

• Altitude = or commonly known as the height (this is needed for the area)
• Median = the midpoint to the opposite side (may not divide the angle of the origin point in half)
• Perpendicular bisector = creates a midpoint on one of the sides and 90 degree angles on both sides of the line
• Angle bisector = divides the origin angle in half, but then doesn’t necessarily divide the opposite side in half

Question 13

How to find number of degrees contained in a polygon

Answer 13

Equation for number of degrees in polygon:
N = # sides
(N - 2 ) 180 = # degrees in polygon

Question 14

What is the best strategy dealing with circles?

Answer 14

Find the radius and then go from there

Question 15

Circumference of Circle

Answer 15

Circumference = Pi * D
or = 2r * Pi

Question 16

Area of circle

Answer 16

A = Pi * r * r

Question 17

What can we say about two radii in a circle that forms a triangle?

Answer 17

• that it’s isosceles
• 2 sides (radii) are the same and thus has 2 equal angles
• Find one of the angles in this triangle and we can find all of the angles
• it may be an equilateral triangle

Question 18

What can we say about an equilateral triangle that creates a chord and that has two sides that are radii?

Answer 18

• chord side is equal to radius
• all angles are 60 degrees, including the central angle
• the arc opposite of the central angle is 1/6 of the circumference

Question 19

What is the measure of the entire circumference?

Answer 19

360 degrees

Question 20

What can we say about the diameter?

Answer 20

• divides the circle into a semi circle with 180 degree arcs
• it is an essentially a 180 degree angle

Question 21

What can we say about 2 different central angles in a circle that are equal?

Answer 21

• they will have opposite arcs of the same size

- they will also have equal length chords

Question 22

What are inscribed angles and what can we say about them?

Answer 22

• vertices are on the circle
• the sides of an inscribed angle are always two chords that meet at the vertex
• measure of the inscribed angle is half of the measure of the arc it intercepts
  ex) inscribed angle of 40 has an arc at the opposite end of 80 degrees
• any inscribed angle that intercepts a semicircle or diameter for a circle is 90 degrees (right angle)

Question 23

What can we say about a radius and a point of tangency?

Answer 23

• they are perpendicular and form a right angle between them 90 degrees
• you can use a right triangle here to find other facts

Question 24

What is the volume and Surface are of a cube?

Answer 24

V= s^3 (s = side of cube)

SA=6s^2

Question 25

What can you use the pythagorean theorem for in a rectangle?

Answer 25

• find the diagonal length of the rectangle

-

Question 26

What is the volume and total area of a cylindar?

Answer 26

V = pi * r^2 * h

SA = areas of two circles (top and bottom) + area of the sides (think of as rectangle) where height is 1 side of the rectangle and circumference of the circles as the other side 
SA = (2pi * r * h) + (2 * pi * r^2)

Question 27

Does scale factor apply to other geometric shapes other than triangles?

Answer 27

Yes, find the scale factor between two similar shapes and then you can find the areas of both along with the perimeters

Question 28

How do you find the arclength of a circle?

Answer 28

• find the radius in order to find the circumference of the entire circle
• know the entire angle of the complete circle is 360
• Thus, if you know the central angle and the radius, you can find the circumference and the arc length over the circumference is the same proportion of the central angle over 360

arclength / 2pi *r = angle / 360

Question 29

How do you find the area of a slice?

Answer 29

Same thinking and approach as finding the length of an arc:

area of sector / pi * r^2 = angle / 360

Question 30

What is the scale factor for volume of similar shapes?

Answer 30

k is the scale factor , for volumes k^3

Question 31

What should you look for when there are two or more triangles involved and parallel lines are present?

Answer 31

Look for similar triangles

Question 32

Polygon must have:

Answer 32

• Equalateral sides

- Equiangular - all angles are equal

Question 33

How do you find the areas of quadrilaterals?

Answer 33

A = B * H

Question 34

How do you find the area of trapezoids?

Answer 34

Finding the average of the bases and multiplying by the height:

A = ((b1 + b2) / 2) * h

Question 35

What are the properties of squares?

Answer 35

-squares are the most elite quadrilaterals (highest number of special properties)
- square is a rectangle
- square is a rhombus
- square is a parallelogram
If a shape is a square, we are given a ton of information
BUT it is very hard to prove something is a square

Question 36

What are the properties of a rectangle?

Answer 36

• all angles are equal to 90

- all diagonals are congruent or equal

Question 37

What are the properties of a rhombuses?

Answer 37

These are equilateral quadrilaterals - a quad with 4 equal sides
Rhombuses are Parallelograms
properties (in addition to parallelogram properties):
- all 4 sides are equal
- diagonals are perpendicular

Question 38

What are the properties of parallelograms?

Answer 38

• opposite sides are parallel
• opposite sides are equal
• opposite angles are equal
• the diagonals bisect each other - midpoint of each
• sum of the 4 interior angles is 360 degrees
  diagonal is where they intersect
  If any one of the above is true, than the rest are true. If any one of these is false, the rest are false!