Math 2 Honors Final Exam Review

Circle

A circle of radius r with center O is the set of all points that are a distance of r units from O

Line Segment

A set of points on a line with two endpoints

Parallel

Two lines that don’t intersect are called parallel. We also call segments parallel if they extend into parallel lines

Conjecture

A reasonable guess that you are trying to either prove or disprove

Perpendicular Bisector

A line through the midpoint of the segment that’s perpendcular to it

Inscribed Polygon in a Circle

A polygon is inscribed in a circle if it fits inside the circle and every vertex of the polygon is on the circle

Inscribed Circle in a Polygon

A circle is inscribed in a polygon if it fits inside the polygon and every side of the polygon is tangent to the circle

Angle Bisector

A line through the vertex of an angle that divides it into two equal angles

Regular Polygon

A polygon where all of the sides are conguent and all the angles are congruent

Tessellation

An arrangement of figures that covers the entire plane without gaps or overlaps

Assertion

A statement you think is true but have not yet proven

Theorem

A statement that has been proved mathamatically

Image

If a transformation takes A to A’, than A is the original and A’ is the image

Congruent

A figure is congruent to another if there is a sequence of translations, rotations, and reflections that takes the first figure onto the second

Reflection

A reflection is defined using a line. It takes a point to another point that is the same distance from the given line, is on the other side of the given line, and so that the segment from the original point to the image is perpendicular to the given line

Directed Line Segment

A line segment with an arrow at one end specifying a direction

Translation

A translation is defined using a directed line segment. It takes a point to another point so that the directed line segment from the original point to the image is parallel to the given line segment and has the same length and direction

Parallel Line Translations

Translations take lines to parallel lines or to themselves

Rotation

A rotation has a center and a directed angle. It takes a point to another point on the circle through the original point with the given center. The 2 radii to the original point and the image make the given angle

Symmetry

A figure has symmetry if there is a rigid transformation which takes it onto itself (not counting a transformation that leaves every point where it is)

Line of Symmetry

A line of symmetry for a figure is a line such that reflection across the line takes the figure onto itself

Reflection Symmetry

A figure has reflection symmetry if there is a reflection that takes the figure to itself

Rotational Symmetry

A figure has rotation symmetry if there is a roation that takes the figure onto itself (we don’t count rotations using anges such as 0 degrees and 360 degrees that leave every point on the figure where it is)

Vertical Angles Theorem

2 intersecting lines from vertical angles which are congruent

Converse

Converse of a conditional statement is formed by swapping the hypothesis and the conclusion. P → Q, converse is Q → P

Alternate Interior Angles Theorem

If 2 parallel lines are cut by a transversal, then alternate interior angles are congruent

Converse of Alternate Interior Angles Theorem

If 2 lines are cut by a transversal and alternate interior angles are congruent, then the lines have to be paralel

Corresponding Angles Theorem

If 2 parallel lines are cut by a transversal, the corresponding angles are congruent

Converse of Corresponding Angles Theorem

If 2 lines are cut by a transversal and corresponding angles are congruent, then the lines have to be parallel

Triangle Angle Sum Theorem

All the angles in a triangle add up to 180 degrees

Corresponding Parts

For a rigid transformation that takes one figure onto another, a part of the first figure and its image in the second figure are called corresponding parts

CPCFC

Corresponding parts of congruent figures are congruent

CPCTC

Corresponding parts of congruent triangles are congruent

Segment congruence

If two segments have the same length then they are congruent

Auxiliary line

An extra line drawn in a figure to reveal a hiddn structure

Side-Angle-Side Triangle Congruence Theorem (SAS)

In two triangles, if two pairs of congruent corresponding sides and the pair of corresponding angles between the sides are congruent, then the two triangles are congruent

Isosceles Triangle Theorem (ITT)

In an isosceles triangle, the base angles are congruent

Parallelogram

A quadrilateral with opposite sides parallel

Angle-Side-Angle Triangle Congruence Theorem (ASA)

In two triangles, if two pairs of corresponding angles and their pair of corresponding sidesbetween the angles are congruent, then the triangles must be congruent

Parallelogram Property

In parallelograms opposite sides are conguent, opposite angles are congruent, consecutive angles are supplementary, and diagonals bisect each other

Points on a Perpendicular Bisector

If P is a point on the perpendicular bisector of AB, then the distance from P to A is the same as the distance from P to B

Converse of points on a perpendicular bisector

Point C must be the perpendicular bisector of AB, if C is the same distance from A as it is from B

Side-Side-Side Triangle Conguence Theorem (SSS)

In two triangles, if all 3 pairs of corresponding sides are congruent, then the triangles must be congruent

Parallelogram Property

A parallelogram with at least one right angle must be a rectangle. A parallelogram with two adjacent sides congruent must be a rhombus

Rectangle

A quadrilateral with 4 right angles

Rhombus

A quadrilateal with 4 congruent sides

Scale Factor

The factor by which every length in an original figure is increased or decreased when you make a scaled copy

Dilation

A dilation with center P and positive scale factor K takes a point A along the ray PA to another point whose distance is K times farther away from P than A is

Dilations of Lines

A dilation takes a line not passing through the center of dilation to a parallel line, and leaves a line passing through the center unchanged

Parallel Side Splitter (PSS)

If a line splits 2 sides of the triangle proportionally, that line is parallel to the third side of the triangle

Similar

one figure is similar to another if there is a sequence of rigid motions and dilations that takes the first figure so that it fits exactly over the second

CPSTP

Corresponding parts of similar triangles are proportional

Circle similarity

all circles are similar

Angle-Angle Triangle Similarity Theorem (AA)

If 2 triangles have 2 corresponding angles congruent, the 2 triangles are similar

Side-Angle-Side Triangle Similarity (SAS)

In 2 triangles, if 2 pairs of corresponding sides are proportional with the angle in between also congruent, the two triangles are similar

Side-Side-Side Triangle Similarity (SSS)

If we have 2 triangles with all 3 pairs of corresponding sides proportional, the two triangles are similar

Altitude in a triangle

A line segment drawn from a vertex perpendicular to the opposite side

Altitude in a right triangle

In a right triangle, the altitude drawn to the hypotenuse creates 3 similar triangles

Pythagorean Theorem

Pythagorean theorem defines the relationship between the sides in a right triangle

Complementary

two angles are complementary to each other if their measure add up to 90 degrees. The two acute angles in a right triangle are complementary to each other

Trigonometric Ratio

Sine, cosine, and tangent are called trigonometric ratios

Sine

the sine of an acute angle in a right triangle is the ratio (quotient) of the length of the opposite leg to the length of the hypotenuse

Cosine

The cosine of an acute angle in a right triangle is the ratio (quotient) of the length of the adjacent leg to the length of the hypotenuse

Tangent

The tangent of an acute angle in a right triangle is the ratio (quotient) of the length of the opposite leg to the length of the adjacent leg

Arc-sine (inverse sine)

The arcsine of a number between 0 and 1 is the acute angle whose sine (ratio) is that number

Arc-cosine (inverse cosine)

The arccosine of a number between 0 and 1 is the acute angle whose cosine (ratio) is that number

arc-tangent (inverse tangent)

The arctangent of a positive number is the acute angle whose tangent is that number

equation of a circle

a circle with center (h,k) and radius r has equation : (x-h)² + (y-k)² = r²

Parabola

the set of points that are equidistant from a given point, called the focus, and a given line, called the directrix

Directrix

the line that with the focus defines a parabolaFocu

Focus

The point that together with the directrix defines a parabola

Point-Slope Form

The form of an equation for a line with slope m through the point (h,k) written as y-k = m(x-h)

Parallel Lines Slopes

The slopes of parallel lines are the same

Perpendicular Line Slopes

The slopes of perpendicular lines are opposite reciprocals

Median

A line from a vertex of a triangle to the midpoint of the opposite side. Each dashed line in the image is a median

Arc

The part of a circle lying between two points on the circle

Central Angle

An angle formed by two rays whose endpoints are the center of a circle

Chord

A chord of a circle is a line segment both of whose endpoints are on the circle

Inscribed Angle

An angle formed by two chords in a circle that shape an endpoint

Inscribed Angle Theorem

An inscribed angle is half the measure of the arc is intercepts

Tangent line

A line is tangent to a circle if the line intersects the circle at exactly one point

Line tangent to circle theorem

A line is tangent to a circle if and only if it is perpendicular to the radius drawn to the point of tangency

Circumscribed

We say a polygon is circumscribed by a circle if it fits inside the circle and every vertex of the polygon is on the circle

Cyclic Quadrilateral

A quadrilateral whose vertices all lie on the same circle

Circumcenter

The circumcenter of a triangle is the intersection of all three perpendicular bisectors of the triangle’s sides. It is the center of the triangle’s circumscribed circle

Triangle’s Circumcenter Theorem

The perpendicular bisector of a triangle’s sides intersect at a single point which is equidistant from all 3 vertices

Incenter

The intersection of all three of the triangle’s angle bisectors. it is the center of the triangle’s inscribed circle

Triangle incenter theorem

All 3 angle bisectors in a triangle meet at a single point called the triangles incenter

Sector

the region inside a circle lying between two radii of the circle

How to calculate area of a sector

angle/360 x pi r²

How to calculate length of an arc

area/360 × 2 pi r

Radian

The radian measure of an angle whose vertex is at the center of a circle is the ratio between the length of the arc defined by the angle and the radius of the circle

Chance experiment

Something you can do over and over again, and you don’t know what will happen each time

Event

A set of one or more outcomes in a chance experiment

Outcome

One of the things that can happen when you do a chance experiment

Probability

The probability of a chance event is a number from 0 to 1 that expresses the likelihood of the event occurring, with 0 meaning it will never occur and 1 meaning it will always occur

Sample space

List of every possible outcome for a chance experiment

Addition Rule

The addition rule states that given events A and B, the probability of either A or B is given by P(A or B) = P(A) + P(B) - P(A and B)

Dependent events

Two events from the same experiment for which the probability of one even depends on whether the other event happens