Semester II Final Review - Dilation, Similarity, Right Triangles, and Circles

Vocabulary-style flashcards covering standard geometry topics: dilations, triangle similarity (SSS, SAS), special right triangles (30-60-90, 45-45-90), trigonometric ratios (Sine, Cosine, Tangent), coordinate circle equations, and circle properties (area, circumference, sector area, arc length).

Reduction Scale Factor ($k$)

The ratio that occurs when the scale factor is less than $1$ ($k < 1$), calculated as $\frac{\text{Small}}{\text{Big}}$, such as $\frac{8}{12} = \frac{2}{3}$.

Dilation $D_{1.5}(X(3,0), Y(4, 2), Z(6,-2))$ Result

The transformed coordinates $X'(4.5,0)$, $Y'(6,3)$, and $Z'(9,-3)$ obtained by multiplying the original coordinates by the scale factor $1.5$.

SSS Similarity Property

A similarity condition where all corresponding sides are proportional, such as $\Delta OPQ \sim \Delta LNM$ when ratios equal $2$ ($\frac{28}{14} = 2$, $\frac{28}{14} = 2$, $\frac{20}{10} = 2$).

Pythagorean Theorem

The formula $a^2 + b^2 = c^2$ used to find missing sides of right triangles, as seen in the calculation $12^2 + 9^2 = 225$, resulting in $x = 15$.

$30^{\circ}-60^{\circ}-90^{\circ}$ Triangle: Short Leg

The side of the triangle calculated as $\frac{\text{hypotenuse}}{2}$.

$30^{\circ}-60^{\circ}-90^{\circ}$ Triangle: Long Leg

The side of the triangle calculated by multiplying the short leg by $\sqrt{3}$.

$45^{\circ}-45^{\circ}-90^{\circ}$ Triangle: Hypotenuse

The side opposite the right angle, which equals $\text{leg} \times \sqrt{2}$, for example, $13\sqrt{2} = \sqrt{18}$.

SOH-CAH-TOA: Sine Ratio

The trigonometric ratio defined as $\sin(M) = \frac{\text{Opposite}}{\text{Hypotenuse}}$, such as $\frac{\sqrt{119}}{12}$ or $\frac{2\sqrt{3}}{4}$ (which simplifies to $\frac{\sqrt{3}}{2}$).

SOH-CAH-TOA: Cosine Ratio

The trigonometric ratio defined as $\cos(M) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$, such as $\cos(T) = \frac{6}{12} = \frac{1}{2}$.

SOH-CAH-TOA: Tangent Ratio

The trigonometric ratio defined as $\tan(M) = \frac{\text{Opposite}}{\text{Adjacent}}$, such as $\frac{\sqrt{119}}{5}$ or $\tan(48) = \frac{x}{5}$.

Angle of Elevation

The upward angle from the horizontal to a point, such as the $28^{\circ}$ angle from the roof of a shorter building to the top of a taller building.

Angle of Depression

The downward angle from the horizontal to a point, such as the $42^{\circ}$ angle from the roof of a shorter building to the bottom of the taller building.

Standard Equation of a Circle

The coordinate plane formula $(x-h)^2 + (y-k)^2 = r^2$, where $(h, k)$ represents the center and $r$ represents the radius.

Circle $(x+3)^2 + (y-2)^2 = 36$ Parameters

A circle with a center at $(-3, 2)$ and a radius of $6$.

Area of a Circle Formula

$A = \pi r^2$, exemplified by a circle with radius $13\,cm$ having an area of $169\pi\,cm^2$.

Circumference of a Circle Formula

$C = 2\pi r$, exemplified by a circle with radius $13\,cm$ having a circumference of $26\pi\,cm$.

Arc Length Formula

The formula used to find a portion of the circumference, calculated as $\frac{\text{arc measure}}{360} \times 2\pi r$, such as $\frac{315}{360} \times 26\pi$.

Area of a Sector Formula

The formula used to find a portion of the circle's area, calculated as $\frac{\text{arc measure}}{360} \times \pi r^2$.

Inscribed Arc Property

The geometric relationship where the measure of the intercepted arc is twice the measure of its inscribed angle (e.g., $m WV = 42 \times 2 = 184^{\circ}$ or $m DC = 61 \times 2 = 122^{\circ}$).