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