Geometry 9th Finals

Finals Notecards

What is slope (m)?

The rate of change between two points showing how much the y-value changes per 1-unit change in the x-value. It represents the steepness and direction of a straight line.

What is the formula to calculate slope (m) from two coordinate points?

m = (y2 - y1) / (x2 - x1)

What is the most critical algebraic rule to remember when using the slope formula?

Order matters. Whichever coordinate point's y-value you start with on top (y2), you must start with its corresponding x-value on the bottom (x2).

What is slope-intercept form?

y = mx + b, where m is the slope and b is the y-value of the y-intercept.

How should a y-intercept (b) always be written as a coordinate point?

As (0, b), representing the point where the line crosses the y-axis when the value of x is exactly 0.

What are the step-by-step instructions to find a linear equation given two coordinate points?

1. Calculate the slope (m) using the slope formula. 2. Plug the slope and one coordinate point into point-slope form: y - y1 = m(x - x1). 3. Distribute the slope and isolate y to convert it into slope-intercept form (y = mx + b).

What does a slope of 0 mean geometrically and algebraically?

A completely horizontal line written as y = b. The y-value stays perfectly constant no matter how much the x-value changes.

What does an undefined slope mean geometrically and algebraically?

A completely vertical line written as x = c. The x-value stays perfectly constant, resulting in a run of 0. It is undefined because you cannot divide by zero.

How do you know if two lines are parallel based on their equations?

They have the exact same slope (m1 = m2) but entirely different y-intercepts (b1 != b2), meaning they will never intersect.

How do you know if two lines are perpendicular based on their equations?

Their slopes are negative reciprocals of each other, meaning their product equals -1 (m1 * m2 = -1). They intersect to form a perfect 90 degree angle.

What is point-slope form and what is its equation?

y - y1 = m(x - x1). It is a foundational linear equation used to construct a line when you are given a slope (m) and any random coordinate point (x1, y1).

What algebraic trap must you watch out for in point-slope form y - y1 = m(x - x1)?

Negative signs. If your point has a negative coordinate, subtracting a negative changes the sign to addition. Example: Given point (-3, 4), the equation side becomes (x - (-3)) which simplifies to (x + 3).

When is it best to use point-slope form versus slope-intercept form?

Use point-slope form when given a generic point and a slope. Use slope-intercept form when you explicitly know the starting y-intercept (0, b) or need to graph a line quickly.

What is standard form of a linear equation?

Ax + By = C

What are the strict formatting rules for standard form (Ax + By = C)?

A, B, and C must be integers (no fractions or decimals), and the leading coefficient A must be positive.

How do you convert an equation from slope-intercept form (y = mx + b) to standard form (Ax + By = C)?

1. Move the x term to the same side as y using inverse operations. 2. If the x term coefficient is negative, multiply the entire equation by -1. 3. If there are any fractions, multiply every term in the entire equation by the common denominator to clear them.

What is the distance formula?

d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

What geometric theorem is the distance formula derived from?

The Pythagorean theorem (a^2 + b^2 = c^2), where the horizontal leg is a = (x2 - x1) and the vertical leg is b = (y2 - y1).

What is the midpoint formula?

Midpoint = ((x1 + x2)/2, (y1 + y2)/2)

What does the midpoint formula fundamentally calculate, and what is its output?

It calculates the exact average of the x-coordinates and the average of the y-coordinates. Its final output is always a single coordinate point (x, y), not a distance.

When do you use the midpoint formula versus the distance formula?

Use distance to find lengths, perimeters, or dimensions. Use midpoint to find centers, bisectors, or halfway markers.

What is a system of linear equations?

A collection of two or more equations containing common variables. Solving the system means finding the exact (x, y) coordinate point where they intersect and make both equations true simultaneously.

What are the three possible solution outcomes for a system of equations?

1. One solution (lines intersect at one coordinate point). 2. No solution (lines are parallel and never touch). 3. Infinitely many solutions (equations represent the exact same line stacked on top of each other).

What algebraic result happens when solving a system with no solution versus infinite solutions?

No solution leaves a false statement (like 0 = 7) because variables cancel. Infinite solutions leaves a true statement (like 0 = 0) because variables cancel.

What is the substitution method for solving systems?

An algebraic method where you isolate one variable in one equation, and then plug that entire expression into the other equation to solve for the remaining variable.

When is it most efficient to use substitution to solve a system?

When one of the variables in either equation is already isolated, or has a coefficient of 1 (like y = 3x - 5 or x + 4y = 12).

What is the elimination method for solving systems?

An algebraic method where you multiply one or both equations by constants so that one variable has opposite coefficients. Adding the equations together then cancels that variable out entirely.

When is it most efficient to use elimination to solve a system?

When both equations are already lined up in standard form (Ax + By = C) and coefficients can be easily matched and canceled out.

What does factoring a quadratic trinomial mean?

Breaking down a standard quadratic expression (ax^2 + bx + c) into its foundational building blocks: two multiplied binomials. It is the reverse process of FOIL.

How do you factor a quadratic expression when a = 1 (x^2 + bx + c)?

Find two integers that multiply together to equal the constant term c, and simultaneously add together to equal the linear coefficient b. Then write them as (x + first number)(x + second number).

What are the step-by-step instructions to factor a quadratic using the AC method when a > 1 (ax^2 + bx + c)?

1. Multiply a * c. 2. Find two integers that multiply to ac and add to b. 3. Rewrite the middle term (bx) as the sum of those two numbers. 4. Split the expression down the center and factor by grouping.

When does factoring a quadratic over integers fail?

When no two integers exist that can satisfy the multiply to c and add to b rule, meaning the roots are either irrational or complex/imaginary numbers.

What is the quadratic formula?

x = (-b +- sqrt(b^2 - 4ac)) / 2a

What initial setup must occur before you can extract a, b, and c for the quadratic formula?

The entire quadratic equation must be moved to one side using inverse operations so that it is explicitly set equal to zero (ax^2 + bx + c = 0).

What is the discriminant of a quadratic equation?

The specific algebraic expression found underneath the square root radical in the quadratic formula: D = b^2 - 4ac.

What does a positive discriminant (b^2 - 4ac > 0) tell you about a quadratic graph and its solutions?

There are two distinct, real number solutions. Geometrically, the parabola crosses the x-axis exactly twice.

What does a discriminant of zero (b^2 - 4ac = 0) tell you about a quadratic graph and its solutions?

There is exactly one real number solution (a repeating root). Geometrically, the vertex of the parabola rests exactly on top of the x-axis.

What does a negative discriminant (b^2 - 4ac < 0) tell you about a quadratic graph and its solutions?

There are zero real solutions (two complex/imaginary solutions). Geometrically, the parabola floats completely above or below the x-axis and never touches it.

What does the mnemonic device SOHCAHTOA stand for?

SOH: sin = opposite/hypotenuse

What is the exact step-by-step process for setting up a right triangle trigonometry problem?

1. Identify the reference angle. 2. Label the three sides relative to the angle: Hypotenuse (H), Opposite (O), and Adjacent (A). 3. Choose the correct trig ratio based on which side length you know and which one you need to find. 4. Write the equation and solve for the unknown.

How do you identify the Hypotenuse (H) of a right triangle?

It is always the longest side of a right triangle and is located directly across from the fixed 90 degree right angle. It never changes based on the reference angle.

How do you distinguish between the Opposite (O) side and the Adjacent (A) side?

The Opposite side is directly across the triangle, completely untouched by the reference angle. The Adjacent side is the leg directly next to the reference angle that helps form the angle alongside the hypotenuse.

When solving a trig problem, how do you solve for a variable when it is trapped inside the angle position, such as cos(x) = 4/5?

Take the inverse trigonometric function of both sides to isolate x. Example: x = cos^-1(4/5).

What is the standard equation of a circle?

(x - h)^2 + (y - k)^2 = r^2

What do the variables (h, k) and r represent in the circle equation (x - h)^2 + (y - k)^2 = r^2?

(h, k) represents the exact coordinate point of the circle's center, and r represents the linear length of the radius.

What algebraic sign transformation trap occurs when identifying a circle's center from its equation?

The signs inside the parentheses are inverted due to the minus signs in the formula. Example: (x + 5)^2 + (y - 3)^2 = 49 has a center of (-5, 3), not (5, -3).

How do you find the radius (r) if you are looking at a standard circle equation?

Take the square root of the constant number isolated on the right side of the equal sign. Example: If the equation ends in = 49, the radius is sqrt(49) = 7.

What algebraic method must you use if a circle's equation is expanded out in general form (x^2 + y^2 + Dx + Ey + F = 0)?

Gather the x terms together, gather the y terms together, move the constant to the right side, and complete the square for both variables to factor it back into standard form.

What is an arc length?

The linear distance of a specific section along the curved outer edge of a circle. It represents a fractional piece of the total circumference.

What is the formula for calculating arc length?

Arc Length = (theta / 360) * 2 * pi * r, where theta is the central angle in degrees and r is the radius.

What is a sector area?

The 2D surface space enclosed inside a specific bounded slice of a circle. It represents a fractional piece of the circle's total area.

What is the formula for calculating a sector area?

Sector Area = (theta / 360) * pi * r^2, where theta is the central angle in degrees and r is the radius.

What is the structural difference between a central angle and an inscribed angle?

A central angle has its vertex resting exactly at the circle's center point. An inscribed angle has its vertex resting on the outer edge of the circle, formed by two intersecting chords.

What is the Inscribed Angle Theorem?

The measure of an inscribed angle is always exactly half the degree measure of its intercepted arc. Equation: Inscribed Angle = 0.5 * Arc.

If a central angle and an inscribed angle intercept the exact same arc, what is the relationship between their angle measures?

The inscribed angle is exactly half the measure of the central angle. (Example: If the central angle is 80 degrees, the inscribed angle is 40 degrees).

What is the formula to find the sum of all interior angles inside any convex polygon?

S = (n - 2) * 180, where n represents the total number of sides.

How do you find the measure of a single interior angle inside a regular polygon?

Calculate the total interior sum and divide it by the total number of angles/sides: Single Angle = ((n - 2) * 180) / n.

What is the sum of all exterior angles for any convex polygon?

It is always exactly 360 degrees, completely regardless of how many sides (n) the polygon has.

What does geometric congruence mean for two polygons?

They are entirely identical in both shape and size. All corresponding side lengths are equal, and all corresponding interior angles are equal.

What does geometric similarity mean for two polygons?

They share the identical shape, but are scaled to different sizes. Their corresponding interior angles are perfectly equal, but their corresponding side lengths are proportional (share a matching ratio).

Given the similarity statement triangle ABC is similar to triangle XYZ, what two pieces of information are explicitly revealed?

1. Corresponding angles are equal (A=X, B=Y, C=Z). 2. Corresponding sides are proportional (AB/XY = BC/YZ = AC/XZ).

What is a translation transformation?

A rigid motion that slides every single point of a figure the exact same linear distance in a specified coordinate direction. Notation: (x, y) -> (x + a, y + b).

What is a reflection transformation?

A rigid motion that flips a geometric figure across a specified line of symmetry, creating a mirror image.

What are the coordinate rules for reflecting a point over the x-axis versus the y-axis?

Reflection over x-axis: (x, y) -> (x, -y). Reflection over y-axis: (x, y) -> (-x, y).

What is a rotation transformation?

A rigid motion that turns a shape around a fixed center point (typically the origin, (0,0)) by a specified angle degree counterclockwise.

What are the coordinate transformation rules for counterclockwise rotations of 90 degrees and 180 degrees around the origin?

90 degree rotation: (x, y) -> (-y, x). 180 degree rotation: (x, y) -> (-x, -y).

What is a dilation transformation, and how does it differ from translations, reflections, and rotations?

A dilation scales a figure larger or smaller from a central point using a scale factor k: (x, y) -> (kx, ky). Unlike the other three rigid transformations, dilation alters size, creating a similar shape rather than a congruent one.

What is the Trapezoid Midsegment Theorem?

The midsegment (EF) connecting the non-parallel sides of a trapezoid is parallel to both bases (AB and DC), and its length is exactly equal to the average of the two bases. Formula: EF = (AB + DC) / 2.

Show the step-by-step algebraic solution for: Base AB = 5x - 9, Base DC = x + 3, Midsegment EF = 2x + 2. Find x.

1. Set up formula: 2x + 2 = ((5x - 9) + (x + 3)) / 2 2. Simplify the right side: 2x + 2 = 6x - 6 / 2 3. Eliminate the fraction: 2x + 2 = 3x - 3 4. Solve for x: 2 = x - 3 Add 3 to both sides: 5 = x

What is the area formula for a Triangle?

A = 0.5 * b * h, where b is the base and h is the perpendicular height.

What is the area formula for a Parallelogram?

A = b * h, where b is the base and h is the vertical perpendicular height (never the slanted side length).

What is the area formula for a Trapezoid?

A = 0.5 * (b1 + b2) * h, where b1 and b2 are the parallel bases and h is the perpendicular height separating them.

What is the area formula for a Circle?

A = pi * r^2, where r is the radius.

What is the general volume formula for any standard Prism or Cylinder?

V = B * h, where B represents the calculated 2D surface area of the base shape, and h is the height extending between the bases.

What is the volume formula for a Rectangular Prism?

V = l * w * h, where l = length, w = width, and h = height.

What is the volume formula for a Cylinder?

V = pi * r^2 * h, where r is the circular base radius and h is the cylinder height.

What is the volume formula for a Cone?

V = (1/3) * pi * r^2 * h. It represents exactly one-third of the volume of a cylinder with matching dimensions.

What is the general volume formula for a Pyramid?

V = (1/3) * B * h, where B is the 2D area of the polygon base and h is the perpendicular height from the base to the top apex point.

What is the volume formula for a Sphere?

V = (4/3) * pi * r^3, where r is the radius.

What is the theoretical probability formula for an event A?

P(A) = Number of favorable outcomes / Total number of possible outcomes in the sample space

What is the difference between theoretical and experimental probability?

Theoretical probability predicts what should happen using ideal mathematical structures. Experimental probability calculates what actually happened by measuring the success ratio of real-world trials.

What is the Complement Rule in probability?

P(A') = 1 - P(A). The probability of an event not happening (A') is equal to 1 minus the probability that it does happen.

What is the Addition Rule of probability (used when finding the probability of event A or event B)?

P(A or B) = P(A) + P(B) - P(A and B)

Why must you subtract P(A and B) in the Addition Rule formula?

To prevent double-counting. You must subtract the overlapping elements that belong to both sets simultaneously.

What does it mean if two events are mutually exclusive, and how does it change the Addition Rule?

It means the two events cannot physically occur at the same time, so their overlap is zero (P(A and B) = 0). The formula simplifies down to: P(A or B) = P(A) + P(B).

What is the definition of independent events?

Two events are independent if the occurrence of the first event has absolutely zero influence on the probability or sample space of the second event (like flipping a coin twice).

What is the Multiplication Rule for two independent events (finding the probability of A and B)?

P(A and B) = P(A) * P(B)

What is the definition of dependent events?

Two events are dependent if the outcome of the first event alters the conditional sample space, changing the probability of the second event (like drawing cards from a deck without replacing them).

What is the Multiplication Rule for two dependent events (finding the probability of A and B)?

P(A and B) = P(A) * P(B given A), where P(B given A) represents the conditional probability that event B occurs given that event A has already taken place.