Geometry Key concepts
Median of a trapezoid
The top parallel side plus the other parallel side divided by 2.
Simplify a Radical√
Think of perfect squares that divide into number ( 4,9,16,25,36,49,64...)
Rigid Motion/Isometry
Two figures are congruent if and only if there exists one, or more, rigid motions which will map one figure onto the other. Distance is preserved. Translations, Reflections, and Rotations are Rigid Motions. Dilations are NOT rigid motions.
Glide reflection
a refection combined with a translation.
Pythagorean theorem
a squared plus b squared equals c squared. the hypotenuse is always c.
Reflection over the y axis
change the sign of the x coordinate.
Reflection over the x axis
change the sign of the y coordinate.
Slope formula
y two minus y one divided by x two minus x one. rise over run.
Area of a trapezoid
½ times the height times the sum of the 2 parallel sides.
Area of a rhombus or kite
1/2 the product of the diagonals
Supplementary angles
2 angles when added together that equal 180 degrees.They do not have to be adjacent angles.
Complementary angles
2 angles when added together that equal 90 degrees. They do not have to be adjacent angles.
Circumference of a circle
2 times pi times radius or pi times diameter.
Special Right Triangles√
30-60-90 side opp 30 = 1/2 hypotenuse side opp 60 = 1/2 hypotenuse (√3) 45-45-90 hypotenuse = leg (√2) leg = 1/2 hypotenuse (√2)
Surface area of a sphere
4 times pi times the radius squared.
The longest side of a triangle is opposite the largest angle
AB is the longest side because it is opposite largest angle. (100)
Trapezoid
At least one set of opposite sides parallel.
Isosceles trapezoid
At least one set of opposite sides parallel. Non-parallel sides are congruent. Base angles are congruent. Diagonals are congruent.
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 triangle
Base times the height divided by 2.
Transformations
Dilations are a capital D. Reflections are a lower case r. Rotations are a capital R. Translations are a capital T.
Arc Length
Find fractional part of circumference
Area of a sector
Find fractional part of whole area
Dilation
Multiply both coordinates by the scale factor.
Get rid of radical in denominator
Multiply by the radical in denominator to rationalize
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.
Lengths of intersecting tangents
Tangents to a circle sharing a common vertex are equal. (Birthday Hat Rule)
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.
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.
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.
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.
Alternate interior angles
alternate interior angles are congruent.
Skew Lines
are lines on different planes that never touch.
Parallel lines
are lines on the same plane that never touch.
Collinear points
are points that lie on the same line
Coplanar points
are points that lie on the same plane
Angle bisector
bisects an angle into 2 congruent angles. they meet in a triangle at a point called the incenter.
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.
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.
Corresponding angles
corresponding angles are congruent.
Perimeter
distance around the outside of a polygon.
Intersection of a tangent and a radius
form right angles when they intersect.
Volume of a sphere
four thirds times pi times the radius to the thrid power.
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.
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.
Rotation of 90 degrees
graph the coordinate. rotate the graph 90 degrees counter clockwise and read the new point.
Point
has no length, width, or thickness. It is indentified by a capital letter.
Line
has no thickness, but it extends forever in both directions.
Slopes of parallel lines
have the same slope and a different y intercept.
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.
Secant
intersects a circle in 2 places.
Tangent
intersects a circle in only 1 place.
Central angle
is equal to its intercepted arc.
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.
Angle formed by 2 chords intersecting in a circle
is equal to the sum of the arcs divided by 2.
Inscribed angle
is equal to ½ its intercepted arc.
Volume of a rectangular prism
length times width times height.
Altitude
makes a right angle with the opposite side. they meet in a triangle at a point called the orthocenter.
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
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.
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.
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.
Volume of a cylinder
pi times radius squared times the height.
Area of a circle
pi times radius squared.
Triangles classified by sides
scalene triangles have no equal sides. isosceles triangles have at least 2 equal sides. equilateral triangles have 3 equal sides.
Adjacent angles
share a common vertex, a common side, but not common interior points.
Isosceles triangles
sides opposite congruent angles in triangle are congruent. angles opposite congruent sides are congruent.(Base Angles of an isosceles triangle are congruent.)
Reflection over the line y=x
switch the x and y coordinates.
Sum of the angles in a triangle
the 3 angles of a triangle add up to 180 degrees.
Exterior angle of a triangle
the exterior angle of a triangle equals the sum of the 2 opposite interior angles.
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.
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.
Distance formula
the square root of x two minus x one squared plus y two minus y one squared.
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.
Lengths of an intersecting 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 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.
Vertical angles
vertical angles are congruent.
Equation of a circle
x minus h squared plus y minus k squared equals radius 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.
