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- Brigham Young University
- Geometry

Jake L.

Slope formula given two points

m = rise/run =

change in y/change in x =

y_{1}-y_{2}/x_{1}-x_{2}

2D Polygon Measures to know:

- interior angles
- perimeters
- area

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3D Polygon Measures to know:

- surface area
- volume

Parrallelogram

opposite sides and opposite angles are parallel

AREA=Base * Height

Rhombus

all sides are equal. opposite angles are equal

AREA=(Diagonal 1 * Diagonal 2)/2

Rectangle

all angles are 90 degrees, and opposite sides are equal

AREA=length * width

Square

All angles are 90 degrees. All sides are equal

Both a rectangle and a rhombus

AREA=side*side

Trapezoid

One pair of opposite sides is parallel

AREA=(Base 1 + Base 2) * Height / 2

sum of interior angles of polygon

(n-2) * 180

n= # of sides

area of a triangle

Area = ½b × h

b = base

h = vertical height

b = base

h = vertical height

GMAT Volume Trick:

Surface Area

the SUM of the areas of ALL of the faces

Volume

Length x Width x Height

Deluxe Pythagorean Theorem

For calculating main diagonal of Cube or Rectangular box:

a^{2} + b^{2} + c^{2 }= d^{2}

SQUARE

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In an isosceles right triangle the hypotenuse is

√2 times the length of the leg.

Triangle Inequality Law: **the sum of any two sides of a triangle must be**

the length of one side of a triangle must be **greater than**

the difference between the lengths of the other two sides

difference AND THE sum

Common Right Triangles

- 3-4-5
- 5-12-13
- 8-15-17

Length of diagonal of a Square w/ Side of x

x Rt 2

x, 2x, x √3

Diagonal of a Cube

diag. = s root 3

Triangles are defined as similar if all their ________ are equal and their ____________ are in proportion

- corresponding angles
- corresponding sides

if two similar triangles have corresponding side lengths in ratio a:b, then their areas will be in ratio _________:________

Their areas will be in the ratio a^{2} : b^{2}.

*The lengths being compared do not have to be sides--they can represent heights or perimeters. In fact, the figures do not even have to be triangles. The principle holds true for any similar figure or polygon.

*The lengths being compared do not have to be sides--they can represent heights or perimeters. In fact, the figures do not even have to be triangles. The principle holds true for any similar figure or polygon.

Area of Equilateral Triangle

Circumference of a Circle

C=2π r

Area of Circle

πr^{2}

Central Angle

Angle whose vertex is at the center of a circle and whose sides are radii.

Inscribed angle

An angle whose vertex is on the circle

An inscribed angles is equal to ______ of the arc it intercepts

half

Any inscribed Triangle where one of its sides is the Diameter of the circle is

A right triangle

Volume and surface area of a cylinder

placing those two sides PERPENDICULAR to each other.

Two major line-angle relationships to know for the GMAT:

- The angles formed by any intersecting lines
- The angles formed by parallel lines cut by a transversal

The interior angles formed by intersecting lines form a circle, so the sum of these angles is

360 degrees

Interior angles that combine to form a line sum to 180 degrees

Angles found opposite each other where these two lines intersect are equal

An exterior angle of a triangle is equal in measure to the sum of:

the two non-adjacent (opposite) interior angles of a triangle

Positive Slope

A slope that goes up from left to right

Negative Slope

A slope that goes down from left to right

zero slope

line without any steepness, or a horizontal line

Undefined slope

- vertical line
- zero in denominator

the x-intercept is the point on the line at which

y=0

the y-intercept is the point on the line at which

x=0

The distance between two points in the xy- plane can be found by using?

The Pythagorean theorem

Midpoint between two points

the perpendicular bisector of a line segment ____1____ and _____2______

- forms a 90 degree angle with the segment
- divides the segment exactly in half

the perpendicular bisector has the _____ _____ slope of the line segment it bisects

negative reciprocal

- the products of the two slopes is -1
- the only exception occurs when one line horizontal and the other line is vertical, since vertical lines have undefined slopes

If two lines in a plane intersect in a single point, the coordinates of that point

solve the equations of __both__ lines

number of x-intercepts is determined by the sign of the **discriminant** in the quadratic equation

- >0 = two roots, two x-intercepts
- =0 =1 root, one x-intercept
- <0 = no roots, the parabola never touches the x-axis, no x-intercepts

Discriminant of a quadratic equation

b²–4ac

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