Module 26: Geometry

Question 1

When n lines intersect through a common point, the sum of all the angles created by the lines is _____

Answer 1

360 degrees

Question 2

Parallel Lines Cut by a Transversal (+ how to know if two lines are parallel)

Answer 2

Frequently used in GMAT problems. All acute angles are equal, and all obtuse angles are equal.

The GMAT may disguise the transversal in a Z shape - don’t be fooled - acute & obtuse angles still equal.

GMAT denotes parallel lines with a “ || “ symbol.

Question 3

sum of interior angles of a polygon

Answer 3

sum = (n-2) * 180, where n is the number of sides.

Question 4

sum of interior angles of a triangle

Answer 4

180

Question 5

sum of interior angles of a quadrilateral

Answer 5

360

Question 6

sum of interior angles of a hexagon

Answer 6

720

Question 7

Triangle Angle Properties (4)

Answer 7

• Angles sum to 180 degrees
• Angles correspond to their opposite sides: the largest angle is opposite the longest side. If two sides are equal, their opposite angles are also equal.
• The sum of any two sides of a triangle must be greater than the third side
• One exterior angle of a triangle is equal to the sum of the two remote interior angles (because both are equal to “180 - the nearest interior angle”)

Question 8

sum of exterior angles of a polygon

Answer 8

360 degrees (ALWAYS)**

Note that this statement is only true if we take only one exterior angle per vertex. (If 2 angles per vertex, we would have 720 degrees.)

Question 9

Area of a triangle

Answer 9

A = (Base x Height) / 2

Height always refers to a straight line from the base to the opposite vertex. To draw this, you may have to calculate it using an “altitude” (a line outside the triangle that is perpendicular to the base and goes straight up to the top vertex.)

Question 10

The Triangle Inequality Theorem

Answer 10

1) The DIFFERENCE in lengths of two sides of a triangle is always less than the length of the third side.
2) Two sides of a triangle are always greater than the length of the third side when combined.

Question 11

Triangle classifications: Sides

Answer 11

Scalene - all sides diff lengths
Isoceles - 2 sides same length (2 angles are the same, as well)
Equilateral - 3 sides same length (also means each angle is 60 degrees)

Question 12

Pythagorean Theorem + when to use it

Answer 12

FOR RIGHT TRIANGLES ONLY:

a^2 + b^2 = c^2, where C is the length of the hypotenuse (side across from the right angle)

Question 13

Isosceles Right Triangle - length ratio (+ a shortcut to remember)

Answer 13

leg - leg - hypotenuse
x - x - x(sqrt2)

Isosceles right triangles are 1/2 of a square. So if you’re working with the diagonal of a square and need to figure out side lengths, remember this ratio

Question 14

30 - 60 - 90 triangle: length ratio for opposite sides (+ a shortcut to remember)

Answer 14

30 - 60 - 90

x - x(sqrt3) - 2x

Two 30-60-90 triangles form an equilateral triangle (when joined on the long side)

Question 15

Area of an equilateral triangle

Answer 15

[s^2 * sqrt(3)] / 4, where S is one side of the triangle

Question 16

3 ways to spot similar triangles

Answer 16

1) 2 angles are the same
2) 3 sides are the same
3) One angle matches, AND the sides forming this angle are in the same ratio

Question 17

3 Common RIGHT Triangle Combos (w/ common multiples)

Answer 17

1) 3-4-5, or 9 + 16 = 25
- Look out for common multiples:
- 6-8-10
- 9-12-15
- 12-16-20

2) 5-12-13, or 25 + 144 = 169
- Common multiple: 10-24-26

3) 8-15-17, or 64 + 225 = 289

Question 18

4 Common Quadrilaterals & properties

Answer 18

Parallelogram: opposite sides and opposite angles are equal; the diagonals bisect each other; adjacent angles add up to 180 degrees

Rectangle: All angles are 90 degrees, and opposite sides are equal.

Square: All angles are 90 degrees, all sides are equal

Trapezoid: One pair of sides is parallel, the other side is not.

Question 19

sum of interior angles of a quadrilateral

Answer 19

360

Question 20

Area of a square

Answer 20

A = s*s, where S is the length of one side of the square

Question 21

Area of a parallelogram

Answer 21

A = Base * Height

Question 22

Area of a trapezoid

Answer 22

A = [(Base 1 + Base 2)/2] * Height

Question 23

The diagonal of a rectangle will divide the shape into two 30-60-90 right triangles if _____.

Answer 23

the length and width of the rectangle for the ratio:

x: x(sqrt(3)).

Question 24

If a square, a circle, and a rectangle all have the same perimeter, the ______ will always have the ____ area.

Answer 24

circle will have the GREATEST area, followed by square. Rectangle is smallest

Question 25

If a square, a circle, and a rectangle all have the same area, the _______ will always have the _____ perimeter.

Answer 25

circle will have the SMALLEST perimeter, followed by square. Rectangle is biggest

Question 26

Area of a REGULAR hexagon (2 formulas)

Answer 26

If the distance between two parallel sides is known:

1.5ds, where d = distance and s = side length

If not:

2.6s^2, where S = one side (this is an approximation. Real version: [3(sqrt3)/2] * s^2

Question 27

A regular hexagon can be divided into _________.

Answer 27

six equilateral triangles.

Question 28

Circumference (2 formulas)

Answer 28

1) C = pi x d

2) C = pi x 2r

Question 29

Radius (2 formulas)

Answer 29

r = d/2
c = 2 x pi x r
a = pi x r^2 (harder to use than C)

Question 30

Area of a circle

Answer 30

A = pi x r^2

Question 31

central angle

Answer 31

Any angle with the vertex at the center of the circle. Formed by two radii

Question 32

Three Equivalent Circle Ratios

Answer 32

central angle/360 = arc length/circumference = area of sector/area of circle.

Question 33

Area of a sector of a circle

Answer 33

First, find the area of the entire circle. Then, use the central angle to determine what fraction of the circle is covered by the sector.

60 degree angle: 60/360 = 1/6 of the area of the circle.

Question 34

Inscribed vs. Central Angles

Answer 34

An inscribed angle has a vertex on the circumference. Central angles have a vertex in the center.

An inscribed angle is equal to half of an equivalent central angle (example: 30 degrees vs. 60 degrees) if they share the same end points. (Remember that they may take on different orientations, but still have same end points.)

An inscribed angle is also equal to half of the degree measure of its corresponding arc.

Question 35

Inscribed Triangles + important rule

Answer 35

All vertices must be points on the circumference of the circle.

RULE: If one of the sides of an inscribed triangle is a diameter of the circle, then the triangle must be a right triangle.

Question 36

When an equilateral triangle is inscribed in a circle, the following must be true: (3 things)

Answer 36

• The center of the triangle is also the center of the circle
• Drawing a line segment from the center to a vertex of the triangle would form a radius, AND that line would bisect the triangle’s 60 degree angle into two 30 degree angles
• The circumference of the outer circle is divided into three arcs of equal length

Question 37

When a triangle has 2 points on the circumference of the circle and 1 point in the middle, the following must be true:

Answer 37

• It is an isoceles triangle, and the 2 equal sides are radii.

Question 38

When a circle is inscribed inside an equilateral triangle, the following must be true:

Answer 38

• Drawing line segments from the center of the circle to either base angle of the triangle and to the center of the base forms two smaller 30-60-90 triangles
• The circle touches the triangle at exactly 3 points, each one the midpoint of the side

Question 39

When a triangle is inscribed in a square:

Answer 39

The base & height of the triangle match the sides of the square (so therefore, the base & height of the triangle are the same.)

Question 40

When a rectangle is inscribed in a circle:

Answer 40

The diagonal of the rectangle is also the diameter of the circle.

Question 41

Regular polygon: definition

Answer 41

All angles and sides are equal.

Question 42

When a regular polygon is inscribed in a square:

Answer 42

the polygon will divide the circumference of the circle into N equal arc lengths, where N is the number of sides of the polygon.

Question 43

Area of an outer circular ring

Answer 43

pi(ra^2 - rb^2)

Where ra = radius of the entire shape and rb = radius of the inner circle

Question 44

Volume of a rectangular solid (+ a trick to remember)

Answer 44

length * width * height

If cube: (edge)^3

Recall that you can’t determine how many items can fit in a rectangular 3D space unless you know the dimensions of each item.

Question 45

The longest diagonal line segment that could fit in a rectangular solid or cube (diff formulas for each…)

Answer 45

Rectangular solid: d^2 = l^2 + w^2 + h^2
where D is the line length.

Cube: d = s*sqrt(3), where S is the length of one side.

Question 46

Surface area of a cube

Answer 46

6s^2 (s = length of one side)

Question 47

Surface area of a rectangular solid

Answer 47

2(LW) + 2(LH) + 2(HW)

L = length
W = width
H = height

Question 48

Volume of a cylinder

Answer 48

V = pi x r^2 x h

Question 49

Surface area of a cylinder

Answer 49

2(pi x r^2) + 2(pi x r x h)

Think of it as the two bases plus the side.