Geometry EOC Review and Mathematical Concepts

Lines, Angles, and Logic

Parallel line relationships involve specific angle sums, such as the sum of $\angle 1 + \angle 2 = 80^{\circ}$ in certain configurations. Transversals across parallel lines create congruent Alternate Interior, Alternate Exterior, Corresponding, and Vertical angles, while Consecutive Interior angles are supplementary ($180^{\circ}$).

Conditional statements ($P \rightarrow Q$) are logically analyzed through their Converse ($Q \rightarrow P$), Inverse ($\neg P \rightarrow \neg Q$), and Contrapositive ($\neg Q \rightarrow \neg P$). For the statement "If the quadrilateral is a square, then it is also a rectangle," the conditional and contrapositive are true, while the converse and inverse are false (counterexample: a rectangle with unequal sides).

Coordinate Geometry and Transformations

The slope of a line perpendicular to $y = \frac{2}{3}x - 5$ is the negative reciprocal, $-\frac{3}{2}$. The equation for a line passing through $(-4, 5)$ with this slope is $y = -\frac{3}{2}x - 1$. The midpoint $M$ of segment $PR$ with $P(20, -5)$ and $M(15, -2)$ results in point $R$ having an $x$-coordinate of $10$. The distance between $T(2, 10)$ and $V(18, -18)$ is approximately $32.25$.

Rigid transformations (translations, reflections, and rotations) preserve congruence. For example, $(x, y) \rightarrow (x + 2, y - 1)$ is a translation, and $(x, y) \rightarrow (-x, -y)$ is a $180^{\circ}$ rotation. Non-rigid transformations, like the dilation $(x, y) \rightarrow (2x, 2y)$, result in similar rather than congruent figures.

Triangles and Trigonometry

To form a triangle, the sum of the two shorter sides must exceed the third side; thus, lengths $8, 8, 20$ cannot form a triangle. Triangles are classified as right if $a^2 + b^2 = c^2$, obtuse if $a^2 + b^2 < c^2$, and acute if $a^2 + b^2 > c^2$. Congruence can be proven via $SSS$, $SAS$, $ASA$, and $AAS$. In similar triangles, such as $\Delta FED \sim \Delta VUT$, corresponding angles are congruent and sides are proportional.

Trigonometric ratios are used to find missing dimensions, such as an iceberg height of $209.8\,m$ or a roof pitch angle of $33.7^{\circ}$ (given a $4\,ft$ rise over a $6\,ft$ run). Common ratios include $\sin(C)$, $\cos(A)$, and $\tan(C)$. Points of concurrency include the Centroid (medians), Circumcenter (perpendicular bisectors), Orthocenter (altitudes), and Incenter (angle bisectors).

Circles and Polygons

The standard equation of a circle is $(x - h)^2 + (y - k)^2 = r^2$. For $(x - 4)^2 + (y - 5)^2 = 25$, the center is $(4, 5)$ and the radius is $5$. To transition between circles of different sizes, a dilation is required (e.g., Circle B is a dilation of Circle A by a factor of $\frac{5}{3}$), proving all circles are similar. Related measures include arc length ($15.35$) and inscribed angles where the angle measure is half the intercepted arc (e.g., $\angle SRT = 34^{\circ}$ for a $68^{\circ}$ arc).

Regular polygons have exterior angles equal to $\frac{360}{n}$; a regular hexagon has an exterior angle of $60^{\circ}$. Parallelograms feature congruent opposite angles (e.g., $\angle E = \angle G = 145^{\circ}$ and $\angle F = \angle H = 35^{\circ}$) and diagonals that bisect each other.

Volume and Three-Dimensional Space

Volume calculations include the sphere ($V = \frac{4}{3}\pi r^3$), where a radius of $30\,cm$ yields $113,040\,cm^3$. For a refrigerator, increasing the height by $8\,inches$ while keeping length and width constant adds $8640\,cubic\,inches$ of space. Density is calculated as $\text{Mass} / \text{Volume}$, applied here to a concrete cylinder with a mass of $14,500\,kg$.

Rotating a rectangle about one side forms a cylinder, while rotating a right triangle around a leg forms a cone. A cross-section of a cylinder cut perpendicular to the base through the center is a rectangle (e.g., $6\,cm \times 10\,cm$). When dimensions of a figure are doubled, the area increases by a factor of $4$ ($2^2$) and the perimeter by a factor of $2$.