Oblique Triangles, Bearings, Geometric Transformations, and Quadratic Inequalities Practice Quiz

A set of vocabulary and rule-based flashcards covering oblique triangles, bearings, geometric transformations, and quadratic inequalities based on the practice quiz transcript.

Law of Sines (SSA) formula

$$\frac{a}{\sin(A)} = \frac{b}{\sin(B)}$$

Law of Sines (Ambiguous Case: 0 Triangles)

Occurs when side $a$ is less than height $h$, where $h = b \sin(A)$.

Law of Sines (Ambiguous Case: 2 Triangles)

Occurs when the height $h$ is less than side $a$, and side $a$ is less than side $b$ ($h < a < b$).

Law of Cosines (SAS)

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

Law of Cosines (SSS) for Angle C

$$\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}$$

Area of an Oblique Triangle

$$\text{Area} = 0.5 \times a \times b \times \sin(C)$$

Quadrant Bearing to True Bearing (N 40° E)

In Quadrant I, the true bearing is expressed as a 3-digit number: $040^\circ$.

Quadrant Bearing to True Bearing (S 25° W)

In Quadrant III, calculated as $180^\circ + 25^\circ = 205^\circ$.

Quadrant Bearing to True Bearing (N 55° W)

In Quadrant IV, calculated as $360^\circ - 55^\circ = 305^\circ$.

True Bearing to Quadrant Bearing (135°)

Located in Quadrant II, calculated as $180^\circ - 135^\circ = 45^\circ$ East of South, or $S\,45^\circ\,E$.

Back Bearing Formula

The bearing from Point B back to Point A, calculated by adding $180^\circ$ to the initial true bearing (e.g., $075^\circ + 180^\circ = 255^\circ$).

Translation Vector $T(a, b)$

Applies the transformation $(x, y) \rightarrow (x + a, y + b)$. For example, point $P(-3, 5)$ translated by $T(4, -7)$ becomes $(1, -2)$.

Reflection over the y-axis

A transformation where $(x, y) \rightarrow (-x, y)$.

Reflection over the line $y = x$

A transformation where $(x, y) \rightarrow (y, x)$.

Reflection over the x-axis

A transformation where $(x, y) \rightarrow (x, -y)$.

Rotation 90° counterclockwise (CCW)

A transformation about the origin where $(x, y) \rightarrow (-y, x)$.

Rotation 180°

A transformation about the origin where $(x, y) \rightarrow (-x, -y)$.

Dilation (Scale Factor k)

A transformation centered at the origin where $(x, y) \rightarrow (kx, ky)$.

Quadratic Inequality Solution (e.g., $x^2 - 5x + 6 \le 0$)

Involves factoring $(x - 2)(x - 3) \le 0$ and testing intervals to find the set between roots, such as $[2, 3]$.

Quadratic Inequality Discriminant Property

If the discriminant $b^2 - 4ac < 0$ and the parabola opens upward, the expression is always positive and has no solution for inequalities where the expression must be $< 0$.

Vertex of a Boundary Parabola

The x-coordinate is found using $x = \frac{-b}{2a}$, then substituted into the equation to find the y-coordinate.

Dashed Boundary Line

Used when graphing strict inequalities ($y >$ or $y <$).

Solid Boundary Line

Used when graphing non-strict inequalities ($y \ge$ or $y \le$).

Shading Rules for Inequalities

Shade above the parabola for $y >$ or $y \ge$; shade below for $y <$ or $y \le$.

Test Point Verification

Substituting a point, such as the origin $(0, 0)$, into the inequality to check if the statement is true (e.g., $0 \ge -4$ is true).