Geometry (Common Core) Facts for the Regents Exam

Notes to the Student

• This study guide contains essential information, formulas, and concepts for the Geometry Regents exam.
• It is designed for students but can be used by teachers.
• Memorize and understand the material.
• Complete practice exams, using the study guide as a reference.
• For additional help, visit www.nysmathregentsprep.com for fully explained Regents exam videos.

Notes to the Teacher

• This is the fourth edition, published in spring 2018.
• It includes missing topics from the third edition:
  • Speed and average speed formulas from Algebra 1
  • Coordinate geometry proof properties for quadrilaterals
• Formatting was updated, diagrams were improved, and typos were corrected.
• Photocopy and distribute to students.
• For other Regents level courses, visit www.nysmathregentsprep.com.
• NYS Mathematics Regents Preparation is a non-for-profit organization.
• Consider becoming a patron to support the creation of more material.

Dedication

• The study guide is dedicated to mathematics teachers of Farmingdale High School and other teachers who have provided inspiration.

Copyright Information

• No part of this document can be reproduced or redistributed within a paid setting without written permission from Trevor Clark.
• The study guide can be photocopied and distributed to students for educational purposes.
• The logo, author, or copyright on each page may not be altered.

Polygons – Interior/Exterior Angles

• Sum of Interior Angles: $180(n − 2)$
• Each Interior Angle of a Regular Polygon: $\frac{180(n-2)}{n}$
• Sum of Exterior Angles: $360°$
• Each Exterior Angle: $\frac{360}{n}$

Triangles

Classifying Triangles

Sides:

• Scalene: No congruent sides
• Isosceles: 2 congruent sides
• Equilateral: 3 congruent sides

Angles:

• Acute: All angles are < 90°
• Right: One right angle that is 90°
• Obtuse: One angle that is > 90°
• Equiangular: 3 congruent angles (60°)
• All triangles have 180°

Exterior Angle Theorem:

• The exterior angle is equal to the sum of the two non-adjacent interior angles.

Midsegment:

• A segment that joins two midpoints
  • Always parallel to the third side
  • $\frac{1}{2}$ the length of the third side
  • Splits the triangle into two similar triangles

Coordinate Geometry

• Slope-Intercept Form of a Line: $y = mx + b$ where $m$ is the slope and $b$ is the y-intercept.
• Point-Slope Form of a Line: $y − y<em>1 = m(x − x</em>1)$ where $m$ is the slope, and $x<em>1$ and $y</em>1$ are the values of a given point on the line.
• Slope Formula: $m = \frac{△y}{△x} = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$

Slopes:

• Parallel lines have the same slope
• Perpendicular lines have negative reciprocal slopes
• Collinear points are points that lie on the same line.
• Midpoint Formula: $(\frac{x<em>1+x</em>2}{2} , \frac{y<em>1+y</em>2}{2})$
• Distance Formula: $d = \sqrt{(x<em>2 − x</em>1)^2 + (y<em>2 − y</em>1)^2}$
• Segment Ratios to Partition Line Segments:
  • $\frac{x - x<em>1}{x</em>2 - x} = \frac{a}{b}$
  • $\frac{y - y<em>1}{y</em>2 - y} = \frac{a}{b}$

Isosceles Triangle

• $2 ≅$ sides and $2 ≅$ base angles
• The altitude drawn from the vertex is also the median and angle bisector
• If two sides of a triangle are $\cong$, then the angles opposite those $\cong$ sides are $\cong$

Parallel Lines

• Alternate interior angles are congruent
• Alternate exterior angles are congruent
• Corresponding angles are congruent
• Same-side interior angles are supplementary

Side – Splitter Theorem

• If a line is parallel to a side of a triangle and intersects the other two sides, then this line divides those two sides proportionally.

Triangle Inequality Theorems

• The sum of 2 sides must be greater than the third side
• The difference of 2 sides must be less than the third side
• The longest side of the triangle is opposite the largest angle
• The shortest side of the triangle is opposite the smallest angle

Trigonometry (SOHCAHTOA)

• When solving for a side, use the sin, cos, and tan buttons
• When solving for an angle, use the sin-1, cos-1, and tan-1 buttons
• SOHCAHTOA:
  • $sin = \frac{opposite}{hypotenuse}$
  • $cos = \frac{adjacent}{hypotenuse}$

Cofunctions:

• Sine and Cosine are cofunctions, which are complementary
  • $sin(x) = cos(90° − x)$
  • $cos(x) = sin(90° − x)$
• If $∠A$ and $∠B$ are the acute angles of a right triangle, then $cos A = sin B$

Triangle Congruence Theorems

• Side-Side-Side (SSS)
• Side-Angle-Side (SAS)
• Angle-Side-Angle (ASA)
• Angle-Angle-Side (AAS)
• Hypotenuse-Leg (HL)
• CPCTC – Corresponding Parts of Congruent Triangles are Congruent

Similar Triangle Theorems

• Angle-Angle (AA)
• Side-Angle-Side (SAS)
• Side-Side-Side (SSS)
• Similar figures have congruent angles and proportional sides
• CSSTP - Corresponding Sides of Similar Triangles are in Proportion
• In a proportion, the product of the means equals the product of the extremes

The Pythagorean Theorem

• To find the missing side of any right triangle if two sides are given, use: $a^2 + b^2 = c^2$ where a and b are the legs, and c is the hypotenuse

The Mean Proportional

• Altitude Theorem (SAAS / Heartbeat Method):
  • The altitude is the geometric mean between the 2 segments of the hypotenuse.
  • $\frac{p<em>1}{a} = \frac{a}{p</em>2}$
• Leg Theorem (HYLLS / PSSW):
  • The leg is the geometric mean between the segment it touches and the whole hypotenuse.
  • $\frac{w}{l} = \frac{l}{h}$

Transformational Geometry

Reflection – FLIP

• $r_{x-axis}(x, y) = (x, −y)$
• $r_{y-axis}(x, y) = (−x, y)$
• $r_{y=x}(x, y) = (y, x)$
• $r_{y=-x}(x, y) = (−y, −x)$
• $r_{(0,0)}(x, y) = (−x, −y)$

Rotation – TURN

• $R_{90°}(x, y) = (−y, x)$
• $R_{180°}(x, y) = (−x, −y)$
• $R_{270°}(x, y) = (y, −x)$

Translation – SHIFT/MOVE

• $T_{a,b}(x, y) = (x + a, y + b)$

Dilation – ENLARGEMENT/REDUCTION

• $D_k(x, y) = (k ⋅ x, k ⋅ y)$
  • Dilations create similar figures, where the corresponding sides are in proportion and the corresponding angles are congruent.
  • Dilations are not always rigid motions, since they do not always preserve distance or congruency.

Rigid Motion:

• A type of transformation that preserves distance, congruency, angle measure, size, and shape.

Composition of Transformations

• When you see “∘”, work from right to left.
• $R<em>{90°} ∘ T</em>{3,-4}$
• The example shows a translation to the right by three units and down by four units, followed by a rotation of 90 degrees.

Types of Composition Transformations

• A composition of 2 reflections over 2 parallel lines is equivalent to a translation.
• A composition of 2 reflections over 2 intersecting lines is equivalent to a rotation.
• Rotational Symmetry Theorem
• A regular polygon with $n$ sides always has rotational symmetry, with rotations in increments equal to its central angle of $\frac{360°}{n}$.
• Rotational symmetry is commonly referred to as “mapping the figure onto itself”.

Circles

Completing the Square

• The method of “completing the square” is used when factoring by the basic “Trinomial Method”, or “AM” method cannot be applied to the problem.
• The completing the square method is commonly used in geometry to express a general circle equation in center-radius form.
  *Example: Express the general equation $x^2 + 4x + y^2 − 6y − 12 = 0$ in center-radius form.

Circle Definition:

• A 2-dimensional shape made by drawing a curve that is always the same distance from the center.

Circle Equations

• General/Standard Equation of a Circle: $x^2 + y^2 + Dx + Ey + F = 0$ where D, E, and F are constants.
• Center – Radius Equation of a Circle: $(x − h)^2 + (y − k)^2 = r^2$ where $(h, k)$ is the center and r is the radius.

Review of Factoring

• The order of Factoring:
  • Greatest Common Factor (GCF)
  • Difference of Two Perfect Squares (DOTS)
  • Trinomial/”AM Method” (TRI)
• GCF: $ax + ay = a(x + y)$
• DOTS: $x^2 − y^2 = (x + y)(x − y)$
• TRI: $x^2 − x + -6 = (x + 2)(x − 3)$

Graphing Circles

Steps:

• Determine the center and the radius
• Plot the center on the graph
• Around the center, create four loci points that are equidistant from the center of the circle
• Using a compass or steady freehand, connect all four points
• Label when finished
  *Example: Graph $(x − 2)^2 + (y + 3)^2 = 9$
• The center is the point (2,-3)
• The radius is 3

Steps:

• Determine if the squared terms have a coefficient of 1
• If there is a constant/number on the left side of the equal sign, move that constant to the right side
• Insert “boxes” or “blank spaces” after the linear terms to acquire a perfect-square trinomial
• Take half of the linear term(s) and square the number. Insert this number on both the left and right sides
• Factor using the “trinomial method”
• Write your equation
• $x^2 + 4x + y^2 − 6y − 12 = 0$
• $x^2 + 4x + y^2 − 6y = 12$
• x^2 + 4x + __ + y^2 − 6y + __ = 12 + __ + __
• $x^2 + 4x + 4 + y^2 − 6y + 9 = 12 + 4 + 9$
• $(x + 2)(x + 2) + (y − 3)(y − 3) = 25$
• $(x + 2)^2 + (y − 3)^2 = 25$
• Formula: $(\frac{B}{2})^2$

Angle Relationships in a Circle

• Central Angle: $∠x = arc$
• Inscribed Angle: $∠x = \frac{1}{2} arc$
• Tangent-Chord Angle: $∠x = \frac{1}{2} arc$
• Two Chord Angles: $∠x = \frac{arc 1 + arc 2}{2}$
• A tangent is perpendicular to its radius, forming a 90° angle
• An angle that is inscribed in a semicircle equals 90°
• If a quadrilateral is inscribed in a circle, then its opposite angles = 180°

Segment Relationships in a Circle

• (Part)(Part)=(Part)(Part)
  • $(a)(b) = (c)(d)$
• If AB ∥ CD, then AC arc≅ BD arc
• Parallel chords intercept congruent arcs
  • $(W)(E) = (W)(E)$
• (Whole)(External)=(Whole)(External)
  • $(b)(a) = (c)(d)$
• (Whole)(External)=(Tangent)^2
  • $(W)(E) = (T)^2$
• $(c)(b) = (a)^2$
• If AB ≅ CD, then AB arc ≅ CD arc
• If a diameter/radius is perpendicular to a chord, then the diameter/radius bisects the chord and its arc.
• Big ºig − L ºittle 2

Circles (Cont’d)

Area of a Sector

• $A = \frac{1}{2} r^2θ$ where A is the area of the sector, r is the radius, and θ is an angle in radians.
• $A = \frac{n}{360} πr^2$ where A is the area of the sector, n is the amount of degrees in the central angle, and r is the radius

Sector Length

• $s = r ⋅ θ$ where s is the sector length, r is the radius, and θ is an angle in radians.
• -or-

Cavalieri’s Principle:

• If two solids have the same height and the same cross-sectional area at every level, then the solids have the same volume.

Volume Formulas:

• Prism - $V = (Area of Base) ⋅ (Height)$
• Cone
• Pyramid
• Cylinder
• Sphere

Density Formulas:

• $Density = \frac{(Mass)}{(Volume)}$

Cross Sections:

• a surface or shape that is or would be exposed by making a straight cut through something at one or multiple points.

Quadrilaterals

Quadrilateral

• A quadrilateral is a four-sided polygon

Trapezoid

• at least one pair of parallel sides
• Formula: The length of the median of a trapezoid can be calculated using the following formula:
  • Median = $\frac{1}{2} (base1 + base2)$

Isosceles Trapezoid

• each pair of base angles are congruent
• diagonals are congruent
• one pair of congruent sides (which are the called the legs. These are the non-parallel sides)

Parallelogram

• opposite sides are parallel
• opposite sides are congruent
• opposite angles are congruent
• consecutive angles are supplementary
• diagonals bisect each other

Rectangle

• all angles at its vertices are right angles
• diagonals are congruent

Rhombus

• all sides are congruent
• diagonals are perpendicular
• diagonals bisect opposite angles
• diagonals form four congruent right triangles
• diagonals form two pairs of two congruent isosceles triangles

Square

• diagonals form four congruent isosceles right triangles
• Each figure inherits the properties of its parent

How to prove Quadrilaterals

• To prove that a quadrilateral is a parallelogram, it is sufficient to show any one of these properties:
  • Both pairs of opposite sides are parallel
  • Both pairs of opposite sides are congruent
  • Both pairs of opposite angles are congruent
  • One pair of opposite sides are both parallel and congruent
  • Diagonals bisect each other
• To prove that a parallelogram is a rectangle, it is sufficient to show any one of these:
  • Any one of its angles is a right angle
  • One pair of consecutive angles are congruent
  • Diagonals are congruent
• To prove that a parallelogram is a rhombus, it is sufficient to show any one of these:
  • One pair of consecutive sides are congruent
  • Diagonals are perpendicular
  • Either diagonal is an angle bisector

How to prove Triangles

• To prove that a given triangle is an isosceles triangle, it is sufficient to show that two sides are congruent.
• To prove that a given triangle is an equilateral triangle, it is sufficient to show that all three sides are congruent.
• Remember – if there is a coordinate geometry proof on the regents, devise a plan, write it down, and use the coordinate geometry formulas shown in the “Coordinate Geometry” section of this packet to prove some properties!

Constructions

• Hexagon Inscribed in a Circle
• Angle Bisector
• Centroid – The intersection of 3 medians
• There will be either one or two constructions on the Geometry regents. It is important to understand basic constructions.
• Copy a Line Segment
• Perpendicular Bisector
• Perpendicular Line passing through a Point on the Given Line
• Perpendicular Line passing through a Point NOT on the Given Line
• Equilateral Triangle
• Square Inscribed in a Circle
• Parallel Lines
• Median – A median is drawn to its midpoint