May Geometry Notes | Information

Centroid formula

one third times the vertex plus two thirds times the midpoint

The point located one third of the way from the midpoint to the opposite vertex

the centroid

Partitioning a line segment formula

x equals x1 plus the quantity a over a plus b, times the quantity x2 minus x1; y equals y1 plus the quantity a over a plus b, times the quantity y2 minus y1

A ratio of a to b means the point is blank of the way from point 1 to point 2

a divided by a plus b of the way

To dilate a point with a given center and scale factor

subtract the center from the point to get the difference, multiply the difference by the scale factor, then add the center back

90 degree counterclockwise rotation rule

the point (x, y) becomes (negative y, x)

In a 90 degree counterclockwise rotation, which value gets negated

the original y gets negated and moves to the x position; x keeps its sign and moves to y

180 degree rotation rule

the point (x, y) becomes (negative x, negative y)

270 degree counterclockwise rotation rule

the point (x, y) becomes (y, negative x)

When a figure is dilated by scale factor k, its surface area changes by

multiply the original surface area by k squared

When a figure is dilated by scale factor k, its volume changes by

multiply the original volume by k cubed

Two events A and B are independent if

P of A times P of B equals P of A and B

The condition P of A times P of B equals P of A and B tells you

the events are independent

30-60-90 triangle side ratios

short leg to long leg to hypotenuse equals 1 to square root of 3 to 2

A 30-60-90 triangle is formed by

cutting an equilateral triangle in half down the altitude

45-45-90 triangle side ratios

leg to leg to hypotenuse equals 1 to 1 to square root of 2

A 45-45-90 triangle is formed by

cutting a square along its diagonal

Area of a triangle when you know two sides and the included angle

A equals one half times a times b times sine of C, where C is the angle between sides a and b

The formula one half ab sine C requires

two sides and the angle between them

To find an angle using the ratio of opposite to hypotenuse

theta equals the inverse sine of opposite divided by hypotenuse

Law of Sines

a over sine A equals b over sine B equals c over sine C

Use Law of Sines when

you know two angles and any side, or two sides and a non-included angle

Law of Cosines

a squared equals b squared plus c squared minus 2bc times cosine of A

Use Law of Cosines when

you know two sides and the included angle, or all three sides

The operation between b squared plus c squared and 2bc cosine A in the Law of Cosines

minus; it is always subtraction

When you have all three sides and need an angle

use Law of Cosines rearranged: cosine of A equals b squared plus c squared minus a squared, all over 2bc

Steps to find the inverse of a function

replace f of x with y, swap x and y, solve for y, rewrite as f inverse of x

The graph of an inverse function is

the reflection of the original over the line y equals x

An angle whose vertex lies on the circle

inscribed angle

An inscribed angle equals

half of its intercepted arc

An angle whose vertex is at the center of the circle

central angle

A central angle equals

the full measure of its intercepted arc

An angle whose vertex is outside the circle with both sides tangent to it

circumscribed angle

A circumscribed angle and the central angle intercepting the same arc

are supplementary and add to 180 degrees

Opposite angles of a quadrilateral inscribed in a circle

are supplementary and add to 180 degrees

1 radian in degrees

180 divided by pi, which is approximately 57.3 degrees

A full circle in radians

2 pi

To convert degrees to radians

multiply by pi over 180

To convert radians to degrees

multiply by 180 over pi

Area of a sector using radians

A equals one half r squared times theta

Area of a sector using degrees

A equals theta over 360 times pi r squared

Arc length formula using radians

arc length equals r times theta

Volume of a pyramid

one third times base area times height

Volume of a cone

one third times pi times r squared times height

A pyramid holds blank the volume of a prism with the same base and height

one third

A cone holds blank the volume of a cylinder with the same base and height

one third

Area of a trapezoid

one half times the quantity base 1 plus base 2, times height

Area of an equilateral triangle

the square root of 3 divided by 4, times a squared, where a is the side length

Surface area of a cone

pi times r times l plus pi times r squared, where l is the slant height

Surface area of a regular right pyramid

one half times the perimeter of the base times the slant height, plus the area of the base

1 square foot equals blank square inches

144, because 12 squared equals 144

1 cubic foot equals blank cubic inches

1728, because 12 cubed equals 1728

The altitude to the hypotenuse of a right triangle equals

the square root of the product of the two hypotenuse segments it creates

Each leg of a right triangle equals

the square root of the whole hypotenuse times the hypotenuse segment adjacent to that leg