Geometry Regents Exam Review Flashcards

Circle Equations

• General Equation: x^2 + y^2 + Cx + Dy + E = 0

• Center-Radius Equation: (x-h)^2 + (y-k)^2 = r^2, where (h, k) is the center and r is the radius.

• Completing the square method is used to express a general circle equation in center-radius form.

  1. Make sure the squared terms have a coefficient of 1

  2. Move the constant to the right side of the equation

  3. Insert "boxes" after the linear terms to create a perfect-square trinomial

  4. Take half of the linear term, square it, and add the result to both sides of the equation

  5. Factor using the "trinomial method"

  6. Write the equation in center-radius form

Graphing Circles

1.  Determine the center and radius
2.  Plot the center
3.  Create four points equidistant from the center (loci points)
4.  Connect the points to form the circle

Triangle Congruence Theorems

• SSS (Side-Side-Side)

• SAS (Side-Angle-Side)

• ASA (Angle-Side-Angle)

• AAS (Angle-Angle-Side)

• HL (Hypotenuse-Leg)

• CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Similar Triangle Theorems

• AA (Angle-Angle)

• SAS (Side-Angle-Side)

• SSS (Side-Side-Side)

• Similar figures have congruent angles and proportional sides

• CSSTP (Corresponding Sides of Similar Triangles are in Proportion)

Mean Proportional Theorems

• Altitude Theorem: \frac{S1}{a} = \frac{a}{S2}

• Leg Theorem: \frac{H}{L1} = \frac{L1}{S1} and \frac{H}{L2} = \frac{L2}{S2}

Pythagorean Theorem

• a^2 + b^2 = c^2, where a and b are legs, and c is the hypotenuse.

Trigonometry (SOHCAHTOA)

• sin(\theta) = \frac{opposite}{hypotenuse}

• cos(\theta) = \frac{adjacent}{hypotenuse}

• tan(\theta) = \frac{opposite}{adjacent}

Cofunctions

• sine = cos(90 - \theta)

• cos(\theta) = sin(90 - \theta)

Coordinate Geometry

• Slope-Intercept Form: y = mx + b

• Point-Slope Form: y - y1 = m(x - x1)

• Slope Formula: m = \frac{\Delta y}{\Delta x} = \frac{y2 - y1}{x2 - x1}

• Parallel lines have the same slope

• Perpendicular lines have negative reciprocal slopes

• Midpoint Formula: M = (\frac{x1 + x2}{2}, \frac{y1 + y2}{2})

• Distance Formula: d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}

Triangle Inequality Theorem

• The sum of 2 sides must be greater than the third side.

Quadrilateral Properties

• Trapezoid: At least one pair of parallel sides.

• Isosceles Trapezoid: Each pair of base angles are congruent, diagonals are congruent, and one pair of congruent sides (legs).

• Parallelogram: Opposite sides are parallel and congruent, opposite angles are congruent, consecutive angles are supplementary, and diagonals bisect each other.

• Rectangle: All angles are right angles and diagonals are congruent.

• Rhombus: All sides are congruent, diagonals are perpendicular and bisect opposite angles.

• Square: Combination of rectangle and rhombus properties.

Angle Relationships in a Circle

• Central Angle: angle = arc

• Inscribed Angle: angle = \frac{1}{2} arc

• Tangent-Chord Angle: angle = \frac{1}{2} arc

• Two Chord Angles: angle = \frac{Arc1 + Arc2}{2}

Segment Relationships in a Circle

• (Part)(Part) = (Part)(Part)

• (Whole)(External) = (Whole)(External)

• (Whole)(External) = (Tangent)^2

• If a quadrilateral is inscribed in a circle, then its opposite angles = 180°

• If a diameter/radius is perpendicular to a chord, then the diameter/radius bisects the chord and its arc.

Circle Equations

• General Equation: x^2 + y^2 + Cx + Dy + E = 0

• Center-Radius Equation: (x-h)^2 + (y-k)^2 = r^2, where (h, k) is the center and r is the radius.

• Completing the square method is used to express a general circle equation in center-radius form.

1. Make sure the squared terms have a coefficient of 1

2. Move the constant to the right side of the equation

3. Insert "boxes" after the linear terms to create a perfect-square trinomial

4. Take half of the linear term, square it, and add the result to both sides of the equation

5. Factor using the "trinomial method"

6. Write the equation in center-radius form

Graphing Circles

1.  Determine the center and radius
2.  Plot the center
3.  Create four points equidistant from the center (loci points)
4.  Connect the points to form the circle

Triangle Congruence Theorems

• SSS (Side-Side-Side)

• SAS (Side-Angle-Side)

• ASA (Angle-Side-Angle)

• AAS (Angle-Angle-Side)

• HL (Hypotenuse-Leg)

• CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Similar Triangle Theorems

• AA (Angle-Angle)

• SAS (Side-Angle-Side)

• SSS (Side-Side-Side)

• Similar figures have congruent angles and proportional sides

• CSSTP (Corresponding Sides of Similar Triangles are in Proportion)

Mean Proportional Theorems

• Altitude Theorem: \frac{S1}{a} = \frac{a}{S2}

• Leg Theorem: \frac{H}{L1} = \frac{L1}{S1} and \frac{H}{L2} = \frac{L2}{S2}

Pythagorean Theorem

• a^2 + b^2 = c^2, where a and b are legs, and c is the hypotenuse.

Trigonometry (SOHCAHTOA)

• sin(\theta) = \frac{opposite}{hypotenuse}

• cos(\theta) = \frac{adjacent}{hypotenuse}

• tan(\theta) = \frac{opposite}{adjacent}

Cofunctions

• sine = cos(90 - \theta)

• cos(\theta) = sin(90 - \theta)

Coordinate Geometry

• Slope-Intercept Form: y = mx + b

• Point-Slope Form: y - y1 = m(x - x1)

• Slope Formula: m = \frac{\Delta y}{\Delta x} = \frac{y2 - y1}{x2 - x1}

• Parallel lines have the same slope

• Perpendicular lines have negative reciprocal slopes

• Midpoint Formula: M = (\frac{x1 + x2}{2}, \frac{y1 + y2}{2})

• Distance Formula: d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}

Triangle Inequality Theorem

• The sum of 2 sides must be greater than the third side.

Quadrilateral Properties

• Trapezoid: At least one pair of parallel sides.

• Isosceles Trapezoid: Each pair of base angles are congruent, diagonals are congruent, and one pair of congruent sides (legs).

• Parallelogram: Opposite sides are parallel and congruent, opposite angles are congruent, consecutive angles are supplementary, and diagonals bisect each other.

• Rectangle: All angles are right angles and diagonals are congruent.

• Rhombus: All sides are congruent, diagonals are perpendicular and bisect opposite angles.

• Square: Combination of rectangle and rhombus properties.

Angle Relationships in a Circle

• Central Angle: angle = arc

• Inscribed Angle: angle = \frac{1}{2} arc

• Tangent-Chord Angle: angle = \frac{1}{2} arc

• Two Chord Angles: angle = \frac{Arc1 + Arc2}{2}

Segment Relationships in a Circle

• (Part)(Part) = (Part)(Part)

• (Whole)(External) = (Whole)(External)

• (Whole)(External) = (Tangent)^2

• If a quadrilateral is inscribed in a circle, then its opposite angles = 180°

• If a diameter/radius is perpendicular to a chord, then the diameter/radius bisects the chord and its arc.

Equation of an Ellipse

• Standard Form: \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 (center at (h, k), major axis length 2a, minor axis length 2b

Equation of a Hyperbola

• Standard Form: \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 (horizontal transverse axis) or \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 (vertical transverse axis)

Area of a Triangle

• Using base and height: A = \frac{1}{2}bh

• Using Heron's formula: A = \sqrt{s(s-a)(s-b)(s-c)} where s = \frac{a

Circle Equations

• General Equation: x^2 + y^2 + Cx + Dy + E = 0

• Center-Radius Equation: (x-h)^2 + (y-k)^2 = r^2, where (h, k) is the center and r is the radius.

• Completing the square method is used to express a general circle equation in center-radius form.

1. Make sure the squared terms have a coefficient of 1

2. Move the constant to the right side of the equation

3. Insert "boxes" after the linear terms to create a perfect-square trinomial

4. Take half of the linear term, square it, and add the result to both sides of the equation

5. Factor using the "trinomial method"

6. Write the equation in center-radius form

Graphing Circles

1.  Determine the center and radius
2.  Plot the center
3.  Create four points equidistant from the center (loci points)
4.  Connect the points to form the circle

Triangle Congruence Theorems

• SSS (Side-Side-Side)

• SAS (Side-Angle-Side)

• ASA (Angle-Side-Angle)

• AAS (Angle-Angle-Side)

• HL (Hypotenuse-Leg)

• CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Similar Triangle Theorems

• AA (Angle-Angle)

• SAS (Side-Angle-Side)

• SSS (Side-Side-Side)

• Similar figures have congruent angles and proportional sides

• CSSTP (Corresponding Sides of Similar Triangles are in Proportion)

Mean Proportional Theorems

• Altitude Theorem: \frac{S1}{a} = \frac{a}{S2}

• Leg Theorem: \frac{H}{L1} = \frac{L1}{S1} and \frac{H}{L2} = \frac{L2}{S2}

Pythagorean Theorem

• a^2 + b^2 = c^2, where a and b are legs, and c is the hypotenuse.

Trigonometry (SOHCAHTOA)

• sin(\theta) = \frac{opposite}{hypotenuse}

• cos(\theta) = \frac{adjacent}{hypotenuse}

• tan(\theta) = \frac{opposite}{adjacent}

Cofunctions

• sine = cos(90 - \theta)

• cos(\theta) = sin(90 - \theta)

Coordinate Geometry

• Slope-Intercept Form: y = mx + b

• Point-Slope Form: y - y1 = m(x - x1)

• Slope Formula: m = \frac{\Delta y}{\Delta x} = \frac{y2 - y1}{x2 - x1}

• Parallel lines have the same slope

• Perpendicular lines have negative reciprocal slopes

• Midpoint Formula: M = (\frac{x1 + x2}{2}, \frac{y1 + y2}{2})

• Distance Formula: d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}

Triangle Inequality Theorem

• The sum of 2 sides must be greater than the third side.

Quadrilateral Properties

• Trapezoid: At least one pair of parallel sides.

• Isosceles Trapezoid: Each pair of base angles are congruent, diagonals are congruent, and one pair of congruent sides (legs).

• Parallelogram: Opposite sides are parallel and congruent, opposite angles are congruent, consecutive angles are supplementary, and diagonals bisect each other.

• Rectangle: All angles are right angles and diagonals are congruent.

• Rhombus: All sides are congruent, diagonals are perpendicular and bisect opposite angles.

• Square: Combination of rectangle and rhombus properties.

Angle Relationships in a Circle

• Central Angle: angle = arc

• Inscribed Angle: angle = \frac{1}{2} arc

• Tangent-Chord Angle: angle = \frac{1}{2} arc

• Two Chord Angles: angle = \frac{Arc1 + Arc2}{2}

Segment Relationships in a Circle

• (Part)(Part) = (Part)(Part)

• (Whole)(External) = (Whole)(External)

• (Whole)(External) = (Tangent)^2

• If a quadrilateral is inscribed in a circle, then its opposite angles = 180°

• If a diameter/radius is perpendicular to a chord, then the diameter/radius bisects the chord and its arc.

Equation of an Ellipse

• Standard Form: \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 (center at (h, k), major axis length 2a, minor axis length 2b

Equation of a Hyperbola

• Standard Form: \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 (horizontal transverse axis) or \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 (vertical transverse axis)

Area of a Triangle

• Using base and height: A = \frac{1}{2}bh

• Using Heron's formula: A = \sqrt{s(s-a)(s-b)(s-c)} where s = \frac{a + b + c}{2}

Law of Sines

• In any triangle, the ratio of the length of a side to the sine of its opposite angle is constant.
  \frac{a}{sin(A)} = \frac{b}{sin(B)} = \frac{c}{sin(C)}

Law of Cosines

• Relates the lengths of the sides of a triangle to the cosine of one of its angles.

• Standard Form:

  • a^2 = b^2 + c^2 - 2bc \cdot cos(A)

  • b^2 = a^2 + c^2 - 2ac \cdot cos(B)

  • c^2 = a^2 + b^2 - 2ab \cdot cos(C)

Volume of a Sphere

• The amount of space inside a sphere

• Formula: V = \frac{4}{3} \pi r^3

Surface Area of a Sphere

• The total area that the surface of the sphere occupies

• Formula: A

Circle Equations

• General Equation: x^2 + y^2 + Cx + Dy + E = 0

• Center-Radius Equation: (x-h)^2 + (y-k)^2 = r^2, where (h, k) is the center and r is the radius.

• Completing the square method is used to express a general circle equation in center-radius form.

1. Make sure the squared terms have a coefficient of 1

2. Move the constant to the right side of the equation

3. Insert "boxes" after the linear terms to create a perfect-square trinomial

4. Take half of the linear term, square it, and add the result to both sides of the equation

5. Factor using the "trinomial method"

6. Write the equation in center-radius form

Graphing Circles

1.  Determine the center and radius
2.  Plot the center
3.  Create four points equidistant from the center (loci points)
4.  Connect the points to form the circle

Triangle Congruence Theorems

• SSS (Side-Side-Side)

• SAS (Side-Angle-Side)

• ASA (Angle-Side-Angle)

• AAS (Angle-Angle-Side)

• HL (Hypotenuse-Leg)

• CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Similar Triangle Theorems

• AA (Angle-Angle)

• SAS (Side-Angle-Side)

• SSS (Side-Side-Side)

• Similar figures have congruent angles and proportional sides

• CSSTP (Corresponding Sides of Similar Triangles are in Proportion)

Mean Proportional Theorems

• Altitude Theorem: \frac{S1}{a} = \frac{a}{S2}

• Leg Theorem: \frac{H}{L1} = \frac{L1}{S1} and \frac{H}{L2} = \frac{L2}{S2}

Pythagorean Theorem

• a^2 + b^2 = c^2, where a and b are legs, and c is the hypotenuse.

Trigonometry (SOHCAHTOA)

• sin(\theta) = \frac{opposite}{hypotenuse}

• cos(\theta) = \frac{adjacent}{hypotenuse}

• tan(\theta) = \frac{opposite}{adjacent}

Cofunctions

• sine = cos(90 - \theta)

• cos(\theta) = sin(90 - \theta)

Coordinate Geometry

• Slope-Intercept Form: y = mx + b

• Point-Slope Form: y - y1 = m(x - x1)

• Slope Formula: m = \frac{\Delta y}{\Delta x} = \frac{y2 - y1}{x2 - x1}

• Parallel lines have the same slope

• Perpendicular lines have negative reciprocal slopes

• Midpoint Formula: M = (\frac{x1 + x2}{2}, \frac{y1 + y2}{2})

• Distance Formula: d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}

Triangle Inequality Theorem

• The sum of 2 sides must be greater than the third side.

Quadrilateral Properties

• Trapezoid: At least one pair of parallel sides.

• Isosceles Trapezoid: Each pair of base angles are congruent, diagonals are congruent, and one pair of congruent sides (legs).

• Parallelogram: Opposite sides are parallel and congruent, opposite angles are congruent, consecutive angles are supplementary, and diagonals bisect each other.

• Rectangle: All angles are right angles and diagonals are congruent.

• Rhombus: All sides are congruent, diagonals are perpendicular and bisect opposite angles.

• Square: Combination of rectangle and rhombus properties.

Angle Relationships in a Circle

• Central Angle: angle = arc

• Inscribed Angle: angle = \frac{1}{2} arc

• Tangent-Chord Angle: angle = \frac{1}{2} arc

• Two Chord Angles: angle = \frac{Arc1 + Arc2}{2}

Segment Relationships in a Circle

• (Part)(Part) = (Part)(Part)

• (Whole)(External) = (Whole)(External)

• (Whole)(External) = (Tangent)^2

• If a quadrilateral is inscribed in a circle, then its opposite angles = 180°

• If a diameter/radius is perpendicular to a chord, then the diameter/radius bisects the chord and its arc.

Equation of an Ellipse

• Standard Form: \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 (center at (h, k), major axis length 2a, minor axis length 2b

Equation of a Hyperbola

• Standard Form: \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 (horizontal transverse axis) or \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 (vertical transverse axis)

Area of a Triangle

• Using base and height: A = \frac{1}{2}bh

• Using Heron's formula: A = \sqrt{s(s-a)(s-b)(s-c)} where s = \frac{a + b + c}{2}

Law of Sines

• In any triangle, the ratio of the length of a side to the sine of its opposite angle is constant.
  \frac{a}{sin(A)} = \frac{b}{sin(B)} = \frac{c}{sin(C)}

Law of Cosines

• Relates the lengths of the sides of a triangle to the cosine of one of its angles.

• Standard Form:

  • a^2 = b^2 + c^2 - 2bc \cdot cos(A)

  • b^2 = a^2 + c^2 - 2ac \cdot cos(B)

  • c^2 = a^2 + b^2 - 2ab \cdot cos(C)

Volume of a Sphere

• The amount of space inside a sphere

• Formula: V = \frac{4}{3} \pi r^3

Surface Area of a Sphere

• The total area that the surface of the sphere occupies

• Formula: A = 4 \pi r^2

Distance on the Coordinate Plane

• Distance Formula: d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}

Parabolas

Standard Form

• (y = ax^2 + bx + c)

Vertex Form

• y = a(x-h)^2 + k

  • (h,k)$$ is