geometry/constructions
Center of the dilation not on the line
A dilation maps a line containing the center of dilation to itself and every line not containing the center of dilation to a parallel line. Dilations preserve angle measure
Cosine
Adjacent / hypotenuse
Midpoint
The middle of the point halfway along
Vertical line equations
X = a
Clockwise 180
(-x,-y)
Counterclockwise 180
(-x,-y)
Over origin
(-x,-y)
Over y axis
(-x,y)
Over y = -x
(-y,-x)
Counterclockwise 90
(-y,x)
Clockwise 270
(-y.x)
Over x = #
(2h-x, y)
Over y = #
(x, 2k-y)
Over x axis
(x,-y)
Clockwise 90
(y,-x)
Counterclockwise 270
(y,-x)
Over y = x
(y,x)
Slope formula
(y2 - y1) / (x2 - x1)
Chord
A chord of a circle is a straight line segment whose endpoints both lie on the circle
Rotation
A circular movement. Rotation has a central point that stays fixed and everything else moves around that point in a circle
Common tangent
A common tangent is a line that is tangent to each of two coplanar circles. A common tangent can be tangent either internally or externally. A common internal tangent is a common tangent that intersects the segment that joins the center of the two circles
Center of dilation on the line
A dilation maps a line containing the center of dilation to itself
Altitude
A line at right angles to a side that goes through the opposite corner
Angle Bisector
A line that splits an angle into two equal angles
Perpendicular Bisectors
A line which cuts a line segment into two equal parts at 90°
Tangent
A line which touches a circle or ellipse at just one point. Below, the blue line is a tangent to the circle
Ratio of the sides
A ratio says how much of one thing there is compared to another thing
Regular
A regular polygon has all angles equal and all sides equal
Secant
A secant line, or just secant, is the infinite line extension of a chord
Semi circle
A semicircle is a one-dimensional locus of points that forms half of a circle. The full arc of a semicircle always measures 180°
Rotational symmetry formula
A shape has Rotational Symmetry when it still looks the same after a rotation
Equilateral Triangle
A triangle in which all theee sides are equal
Isosceles Triangle
A triangle that has two sides of equal length
Scalene Triangle
A triangle with all sides of different lengths
Properties of a rectangle
All the properties of a parallelogram , all angles are right angles , and the diagonals are congruent
Properties of a rhombus
All the properties of a parallelogram, all sides are congruent by definition, the diagonals bisect the angles, and the diagonals are perpendicular bisectors of each other
Properties of a square
All the properties of a rhombus, all the properties of a rectangle, all sides are congruent by definition, and all angles are right angles
Tangent chord angle
An Angle formed by a chord and a tangent that intersect on a circle is half the measure of the intercepted arc
Arc
An arc is a portion of the circumference of a circle
Hyp-Leg
Any two right triangles that have a congruent hypotenuse and a corresponding, congruent leg are congruent triangles
Floating angles
Chords crossing , add each arc then divide by 2
Reflection
Every point is the same distance from the central line and the reflection has the same size as the original image
Proving a scalene triangle
If all three sides have different lengths, then the triangle is scalene
Tangent Secant angle
If an angle is formed by a secant and a tangent that intersect in the exterior of a circle, then the measure of the angle is one-half the difference of the measures of its intercepted arcs
Proving triangles similar by SAS
If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles must be similar
Tangent tangent angle
If angle is formed by two intersecting tangents, then the measure of the angle is onehalf the difference of the measures of the intercepted arcs
Proving a parallelogram
If one pair of opposite sides of a quadrilateral are both parallel and congruent, then the quadrilateral is a parallelogram
Proving triangles similar by SSS
If the lengths of the corresponding sides of two triangles are proportional, then the triangles must be similar
Converse of the Isosceles Triangle Theorem
If two angles of a triangle are congruent, then sides opposite those angles are congruent
Proving triangles similar by AA
If two angles of one triangle are congruent to two angles of another, then the triangles must be similar
Isosceles Triangle Theorem
If two sides of a triangle are congruent, then the angles opposite those sides are congruent
Translation
In a translation, every point of the object must be moved in the same direction and for the same distance
Mid segment theorem
In a triangle, the segment joining the midpoints of any two sides will be parallel to the third side and half its length
Scale factor
In two similar geometric figures, the ratio of their corresponding sides is called the scale factor
Inscribed angle
Is an angle formed by two chords in a circle which have a common endpoint
Major arc
Is an arc of a circle having measure greater than or equal to radians
Minor arc
Is an arc of a circle having measure less than or equal to radians
Perpendicular Lines
Lines that are at right angles 90° to each other
Midpoint formula
M= (x1 + x2) / 2 + (y1 + y2) /2
Dilation
Means to make larger or smaller
Tangent
Opposite / adjacent
Sine
Opposite / hypotenuse
Point symmetry
Point Symmetry is when every part has a matching part: the same distance from the central point. but in the opposite direction
Proving a right triangle
Pythagorean theorem
Same Side Interior Angles
Same side interior angles are two angles that are on the same side of the transversal and on the interior of between the two lines
Writing the equation of a median
Slope formula but choose a point
Writing the equation of a perpendicular bisector
Slope formula but choose a point
Writing the equation of an altitude
Slope formula but choose a point
Supplementary Angles
Supplementary when they add up to 180 degrees
Central angle
The angle that forms when two radii meet at the center of a circle
Image
The figure after the transformation
Angle inscribed in a semi circle
The intercepted arc for an angle inscribed in a semi-circle is 180 degrees. Therefore the measure of the angle must be half of 180, or 90 degrees
Secant secant angle
The measure of an angle formed when two secants intersect at a point outside the circle is one-half the difference of the measures of the two intercepted arcs
Median
The middle number in a sorted list of numbers
Pre-image
The original figure
Properties of a parallelogram
The properties are opposite sides are parallel by definition, opposite sides are congruent, opposite angles are congruent, consecutive angles are supplementary, and the diagonals bisect each other
Tangent radius
The radius of a circle is perpendicular to the tangent line through its endpoint on the circle's circumference
Side splitter theorem
The side splitter theorem states that if a line is parallel to a side of a triangle and intersect the other two sides, then this line divides those two sides proportionally
About the origin
The starting point. On a number line it is 0. Also referred to as(0,0)
Rigid motion
There are four types of rigid motions: translation , rotation, reflection, and glide reflection. Translation: In a translation, everything is moved by the same amount and in the same direction. Every translation has a direction and a distance
How to find perimeter
To find the perimeter of a rectangle or square you have to add the lengths of all the four sides
Congruent Triangles
Triangles are congruent when they have exactly the same three sides and exactly the same three angles
Complementary Angles
Two Angles are Complementary when they add up to 90 degrees
Linear Pairs
Two angles that are adjacent and supplementary
Parallel Lines
Two lines on a plane that never meet and they are always the same distance apart
Similar figures
Two shapes are Similar when the only difference is size
Vertical Angles
Vertical Angles are the angles opposite each other when two lines cross and they are always equal
Dilating the equation of a line
When a figure is dilated, a segment (side) of the pre-image that does not pass through the center of dilation will be parallel to its image. The dilation of a line segment is longer or shorter in the ratio given by the scale factor
Collinear Points
When three or more points lie on a straight line
Corresponding Angles
When two lines are crossed by another line, which is called the Transversal, the angles in matching corners are called corresponding angles
Alternate Interior Angles
When two lines are crossed by another line, which is called the Transversal, the pairs of angles on opposite sides of the transversal but inside the two lines are called Alternate Interior Angles
Alternate Exterior Angles
When two lines are crossed by another line, which is called the Transversal, the pairs of angles on opposite sides of the transversal but outside the two lines are called Alternate Exterior Angles
ASA
When we have two triangles where we know two angles and the included side are equal
AAS
When we have two triangles where we know two angles and the non-included side are equal
SAS
When we have two triangles where we know two sides and the included angle are equal
SSS
When we have two triangles with all three sides equal
Horizontal line equations
Y = b
Slope intercept formula
Y = mx + b
Diameter
a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle
Radius
a radius of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is the length of any of them
Auxiliary Line
adding a line or segment to a diagram to help in solving a problem or proving a concept
Angle bisector theorems
the angle bisector theorem is concerned with the relative lengths of the two segments that a triangle's side is divided into by a line that bisects the opposite angle
Remote Interior Angle Theorem
the two angles inside the triangle that do not share a vertex with the exterior angle
Same Side Exterior Angles
when parallel lines are cut by a transversal, same-side exterior angles are formed which are outside of the parallel lines and on the same side of the transversal line
Point slope formula
y-y₁=m(x-x₁)
Proving an equilateral triangle
•A triangle is equilateral if and only if it is equiangular. •Each angle of an equilateral triangle has a degree measure of 60.
Proving a square
•If a quadrilateral has four congruent sides and four right angles, then it's a square •If two consecutive sides of a rectangle are congruent, then it's a square
Proving a rectangle
•If all angles in a quadrilateral are right angles, then it's a rectangle •If the diagonals of a parallelogram are congruent, then it's a rectangle •If a parallelogram contains a right angle, then it's a rectangle
Proving a rhombus
•If all sides of a quadrilateral are congruent, then it's a rhombus •If the diagonals of a quadrilateral bisect all the angles, then it's a rhombus
Proving an isosceles triangle
•isosceles triangle theorem •converse of the isosceles triangle theorem
Equations for parallel lines
•point slope formula •slope intercept formula •y=2x + 1
Equations for perpendicular lines
•point slope formula •slope intercept formula •y=2x + 1