Math Master Study Sheet Review

Vocabulary practice flashcards covering geometry, trigonometry, algebra, statistics, and research methodology concepts from the Math Master Study Sheet.

30-60-90 Triangle

A special right triangle with side length ratios of $x : x\sqrt{3} : 2x$ (or $x/\sqrt{3} : x : 2x/\sqrt{3}$, or $x/2 : x\sqrt{3}/2 : x$), where the side opposite the $60^\circ$ angle always contains the $\sqrt{3}$ factor.

45-45-90 Triangle

A special right triangle with side length ratios of $x : x : x\sqrt{2}$ (or $x/\sqrt{2} : x/\sqrt{2} : x$).

Complementary Angle Trigonometric Rule

$\sin(A) = \cos(B)$ if and only if $A + B = 90^\circ$. When angles are complementary, their sine and cosine values are equal.

Desmos Regression Tool Strategy

A method used when a given function formula has an equal number of unknown parameters and given coordinate points, where points are plugged into a table to solve for missing parameters directly.

Vieta's Formulas (Quadratic)

For the equation $ax^2 + bx + c = 0$, the sum of roots ($\alpha + \beta$) is $-b/a$ and the product of roots ($\alpha\beta$) is $c/a$.

Vieta's Formulas (Cubic)

For the equation $ax^3 + bx^2 + cx + d = 0$, the sum of roots ($\alpha + \beta + \gamma$) is $-b/a$ and the product of roots ($\alpha\beta\gamma$) is $-d/a$.

Base Time Unit Period Strategy

In exponential models featuring fractional powers/exponents, the strategy of setting the full exponent equal to 1 to find the base time unit period ($t$).

Exponent Rules

Core algebraic rules including the Product Rule ($x^m \times x^n = x^{m+n}$), the Power Rule ($(x^m)^n = x^{m \times n}$), and the Radical/Fractional Rule ($x^{a/n} = \sqrt[n]{x^a} = (\sqrt[n]{x})^a$).

Conditions of Solvability: Slope-Intercept Form

For lines $y = m_1 x + c_1$ and $y = m_2 x + c_2$: One Solution occurs when $m_1 \neq m_2$; No Solution occurs when $m_1 = m_2$ and $c_1 \neq c_2$; Infinitely Many Solutions occur when $m_1 = m_2$ and $c_1 = c_2$.

Conditions of Solvability: Standard Form

For lines $a_1 x + b_1 y = c_1$ and $a_2 x + b_2 y = c_2$: One Solution occurs when $a_1/a_2 \neq b_1/b_2$; No Solution occurs when $a_1/a_2 = b_1/b_2 \neq c_1/c_2$; Infinitely Many Solutions occur when $a_1/a_2 = b_1/b_2 = c_1/c_2$.

Quadratic Vertex from Symmetric Points

The property stating that if $f(a) = f(b)$, the x-coordinate of the parabola's vertex is $(a + b) / 2$.

Parallelogram

A quadrilateral whose diagonals bisect each other; when inscribed in a circle, it is always a rectangle.

Rhombus

A quadrilateral with all sides equal, opposite sides parallel, and opposite angles equal. Its diagonals bisect at right angles ($90^\circ$) but are not equal, its area is $(1/2) \times d_1 \times d_2$, and its angle bisectors enclose a rectangle.

Trapezium (Trapezoid) Median

A line segment whose length equals $(1/2) \times (\text{sum of parallel sides}) = (1/2) \times (AB + CD)$; an inscribed trapezium is an isosceles trapezium.

Kite

A quadrilateral featuring two pairs of adjacent congruent sides with diagonals that intersect at right angles ($90^\circ$).

Regular Polygon Interior Angle Sum

The total sum of interior angles in a polygon, calculated as $(n - 2) \times 180^\circ$, where $n$ is the number of sides.

Maximum / Minimum Revenue Strategy

The problem-solving approach using $\text{Revenue} = (\text{Units}) \times (\text{Price})$, testing 1 to 2 change increments to spot an algebraic pattern, generalizing it into a quadratic function, and locating its vertex.

Identical Equations Strategy

The method of expanding and simplifying both sides of equivalent polynomial forms completely to match corresponding coefficients for identical terms.

Mean

A measure of central tendency that is heavily affected and skewed by extreme outliers.

Median

A measure of central tendency that is robust against outliers and must always be recalculated step-by-step when data points are added or removed.

Standard Deviation

A statistical metric that measures spread or dispersion and is directly proportional to range.

Factor Theorem

An algebraic theorem stating that if $(x - k)$ is a factor of $f(x)$, then $f(k) = 0$.

Quartic Difference of Squares

The algebraic factoring pattern defined as $a^4 - b^4 = (a^2 - b^2)(a^2 + b^2) = (a - b)(a + b)(a^2 + b^2)$.

Two-Way Table Conditional Probability Rule

The rule that the condition following the phrase 'Given that' or the group specified in 'If [Group X] is selected at random' forms the denominator of the probability fraction.

Triangle Similarity Theorems

The three geometric criteria used to establish similarity between triangles: SSS (Side-Side-Side), SAS (Side-Angle-Side), and AAA (Angle-Angle-Angle).

Triangle Congruence Theorems

The geometric criteria used to prove congruence between triangles: SAS, SSS, ASA (Angle-Side-Angle), AAS (Angle-Angle-Side), and RHS / HL (Right-Hypotenuse-Leg).

Cause and Effect (Research Methodology)

A research principle stating that sample surveys cannot infer cause-and-effect, whereas experiments can if a control/comparison group is present and random assignment/allocation is performed.

Margin of Error

A statistical value inversely related to sample size (a larger sample yields a smaller margin of error) and directly related to standard deviation.

Multi-Quantity Percentages Strategy

A technique for percentage problems connecting three or more variables where the common reference variable is set to 1 (or 100) to frame equations and solve directly for the final variable.