Geometry Regents Review

Tangent

intersects a circle in only 1 place.

Secant

intersects a circle in 2 places.

Intersection of a tangent and a radius

form right angles when they intersect.

Equation of a circle

x minus h squared plus y minus k squared equals radius squared.

Circumference of a circle

2 times pi times radius or pi times diameter.

Area of a circle

pi times radius squared.

Central angle

is equal to its intercepted arc.

Inscribed angle

is equal to ½ its intercepted arc.

Angle formed by 2 chords intersecting in a circle

is equal to the sum of the arcs divided by 2.

Angle formed by 2 secants

is equal to the major arc minus the minor arc divided by 2.

Angle formed by a secant and a tangent

is equal to the major arc minus the minor arc divided by 2.

Angle formed by two tangents

is equal to the major arc minus the minor arc divided by 2.

Lengths of 2 intersecting chords

part of the first chord times the other part of the first chord equals a part of the second chord times the other part of the second chord.

Lengths of an intersecting diameter and chord that meet at right angles (perpendicular)

if a diameter meets a chord at a right angle (perpendicular), the diameter divides the chord into 2 equal parts.

Lengths of 2 intersecting secants

the whole length of the first secant times the outside length of the first secant equals the whole length of the second secant times the outside length of the second secant.

Lengths of an instersecting secant and tangent

the whole length of the first secant times the outside length of the first secant equals the length of the tangent squared.

Lengths of intersecting tangents

Tangents to a circle sharing a common vertex are equal.

Angles

acute angles are less than 90 degrees. Right angles are 90 degrees. obtuse angles are between 90 and 180 degrees. Straight angles are 180 degrees and reflex angles are greater than 180 degrees.

Adjacent angles

share a common vertex, a common side, but not common interior points.

Complementary angles

2 angles when added together that equal 90 degrees.

They do not have to be adjacent angles.

Supplementary angles

2 angles when added together that equal 180 degrees.They do not have to be adjacent angles.

Vertical angles

vertical angles are congruent.

Alternate interior angles

alternate interior angles are congruent.

Corresponding angles

corresponding angles are congruent.

Sum of the angles in a triangle

the 3 angles of a triangle add up to 180 degrees.

Triangles classified by sides

scalene triangles have no equal sides.

isosceles triangles have at least 2 equal sides.

equilateral triangles have 3 equal sides.

Triangles classified by angles

acute triangles have 3 acute angles.

right triangles have a 90 degree and 2 acute angles.

obtuse triangles have an obtuse and 2 acute angles.

Exterior angle of a triangle

the exterior angle of a triangle equals the sum of the 2 opposite interior angles.

Isosceles triangles

sides opposite congruent angles are congruent.

angles opposite congruent sides are congruent.

Triangle inequality theorem

the sum of 2 sides of a triangle must be greater than the 3rd side.

Mid segment of a triangle

a mid segment connects the midpoint of 2 sides of a triangle and is equal to ½ the side not containing the 2 midpoints.

Median

bisects the opposite side into 2 congruent line segments.

they meet in a triangle at a point called the centroid.

median segments are in a ratio of 2 to 1.

Angle bisector

bisects an angle into 2 congruent angles.

they meet in a triangle at a point called the incenter.

Altitude

makes a right angle with the opposite side.

they meet in a triangle at a point called the orthocenter.

Perpendicular bisector

bisects and makes a right angle with a side of a triangle.They meet in a triangle at a point called the circumcenter.

Similar triangles

angles in similar (∼) triangles are congruent.

sides are in proportion.

angles are in a proportion of one to one. (1:1)

Proving triangles similar

need only 2 angles to be congruent to probe 2 triangles similar.

Proving triangles congruent

can not be angle angle side (A.S.S.) or side side angle(S.S.A.).

C.P.C.T.C.

corresponding parts of congruent triangles are congruent.

Pythagorean theorem

a squared plus b squared equals c squared.

the hypotenuse is always c.

Proving right triangles congruent

hypotenuse leg.

Right triangle ratios

Slope

from left to right. up the hill is positive slope. down the hill is negative slope. a horizontal line has 0 slope and a verical line as an undefined slope.

Point slope form of a line

y minus y one equals slope (m) times x minus x one.

Slope intercept form of a line

y equals slope (m) times x plus the y intercept (b)

Slope formula

y two minus y one divided by x two minus x one.

rise over run.

Distance formula

the square root of x two minus x one squared plus y two minus y one squared.

Midpoint formula

x one plus x two divided by two is the x coordinate.

y one plus y two divided by two is the y coordinate.

Find the endpoint of a line given the midpoint and the other endpoint

the integers added to the coordinates of C to get the coordinates of M are added to the coordinates of M to get the coordinates of D.

Finding the slope and y intercept of a line

y must be positive and isolated on one side of the equation.

Slopes of parallel lines

have the same slope and a different y intercept.

Slopes of perpendicular lines

the slopes are negative reciprocals.

to find the negative reciprocal make the slope a fraction, flip it, and change the sign.

Parallel and perpendicular lines

Axis of symmetry

the equation for axis of symmetry is x equals negative b divided by two times a.

Surface area of a cylinder

2 times pi times radius times the sum of the height and radius.

Volume of a cylinder

pi times radius squared times the height.

Surface area of a rectangular prism

2 times the sum of the length times the width, the length times the height. and the width times the height.

Volume of a rectangular prism

length times width times height.

Volume of a cone

one third times pi times the radius squared times the height.

Volume of a pyramid

one third times the area of the base times the height.

the base could be a triangle, rectangle, or square

Surface area of a sphere

4 times pi times the radius squared.

Volume of a sphere

four thirds times pi times the radius to the thrid power.

Lateral surface area of a cylinder

2 times pi times the radius times the height.

Lateral surface area of a cone

pi times the radius times the slant height.

Polygons

a triangle has 3 sides. a quadrilateral has 4 sides. a pentagon has 5 sides. a hexagon has 6 sides. a octagon has 8 sides. a decagon has 10 sides.

Formula for angles of polygons

the sum of the interior angles equals the number of sides minus 2 times 180 degrees.

the sum of the exterior angles equal 360 degrees.

Perimeter

distance around the outside of a polygon.

Parallelograms

opposite sides are parallel.

opposite sides and angles are congruent.

consecutive angles are supplemental (add to 180).

diagonals bisect each other.

Rectangle

opposite sides are parallel.

opposite sides are congruent.

contains 4 right angles.

diagonals are congruent and bisect each other.

Square

opposite sides are parallel.

all sides are congruent and meet at right angles.

diagonals are congruent and perpendicular(make right angles).

diagonals bisect each other and the angles of the square.

Rhombus

opposite sides are parallel.

all sides are congruent.

diagonals are congruent and perpendicular(make right angles).

diagonals bisect each other and the angles of the rhombus.

Trapezoid

one set of parallel sides.

Median of a trapezoid

the top parallel side plus the other parallel side divided by 2.

Isosceles trapezoid

one set of parallel sides.

non-parallel sides are congruent.

base angles are congruent.

diagonals are congruent.

Area of a triangle

base times the height divided by 2.

Area of a rectangle

base times height or

length times width.

Area of a square

base times height or side squared.

Area of a parallelogram

base times the height

Area of a rhombus

base times the height

Area of a trapezoid

½ times the height times the sum of the 2 parallel sides.

Transformations

dilations are a capital D.

reflections are a lower case r.

rotations are a captital R.

translations are a capital T.

Dilation

multiply the coordinate by the scale factor.

Reflection over the x axis

change the sign of the y coordinate.

Reflection over the y axis

change the sign of the x coordinate.

Reflection over the line y=x

switch the x and y coordinates.

Rotation of 90 degrees

graph the coordinate. rotate the graph 90 degrees counter clockwise and read the new point.

Rotation of 180 degrees

graph the coordinate. rotate the graph 180 degrees counter clockwise and read the point.

Rotation of 270 degrees

graph the coordinate. rotate the graph 270 degrees counter clockwise and read the point.

Translation

add the 2 integers to the orginal coordinate. add the first integer to the x coordinate and the second integer to the y coordinate.

Glide reflection

a refection combined with a translation.

Isometry

the lengths of corresponding line segments remain the same.

Direct isometry

distance is preserved(same). direction is preserved(same)

Opposite isometry

distance is preserved(same). direction is reversed(opposite).

Conjuntions

and (∧) is true only when both are statements are true.

Disjunctions

or (∨) is only false when both statements are false.

Conditionals

if then (→) is only false when the first statment is true and the second statement is false.

Biconditional

if and only if (↔) is true when the two statements have the same truth value.

Logic statements

for the converse switch the two statements.

for the inverse negate the two statements.

for the contrapositive and logically equivalent, switch and negate the two statements.

Plane

has no thickness and extends forever in all directions.

Line

has no thickness, but it extends forever in both directions.