Comprehensive Geometry and Trigonometry Study Guide

Coordinate Geometry, Distance, and Slopes

The midpoint of a line segment is calculated using the average of the coordinates, while the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is determined by the formula $d = \text{distance} = \text{square root of } (x_2-x_1)^2+(y_2-y_1)^2$. Specifically, the transcript provides the example of finding an endpoint when given a midpoint and another endpoint through linear translation: starting at endpoint $(-3, 4)$, the path to midpoint $(1, 7)$ involves an increase of $+4$ for $x$ and $+3$ for $y$. Applying these same increments to the midpoint results in the next endpoint $(5, 10)$. Slopes, denoted as $m$, represent the rise over the run, calculated as $\frac{y_2-y_1}{x_2-x_1}$. Lines that are parallel have the same slope, whereas perpendicular lines have opposite reciprocal slopes (for instance, if one slope is $a/b$, the perpendicular slope is $-b/a$). Equations of lines can be expressed in slope-intercept form, $y=mx+b$, or point-slope form, $(y-y_1)=m(x-x_1)$.

Properties of Polygons and Transversals

For any polygon where $n$ equals the number of sides, the sum of the interior angles is given by the formula $(n-2) \times 180$. Each individual interior angle in a regular polygon is found by $\frac{(n-2) \times 180}{n}$. The sum of the exterior angles for any polygon is always $360^\text{o}$, and each exterior angle of a regular polygon is calculated as $\frac{360^\text{o}}{n}$. In triangle geometry, the isosceles theorem states that a triangle with two congruent sides also has two congruent angles. The triangle inequality theorem defines the range of the third side of a triangle; for example, if two sides are $8$ and $12$, the range for $x$ is $12-8 < x < 12+8$, or $4 < x < 20$. A triangle's area is $\frac{1}{2}bh$. Three medians of a triangle meet at the center (centroid), which divides the median such that the segments maintain a $1:3$ or $2:3$ ratio. In a transversal intersection, the specific angle pairs (such as alternate interior or corresponding) determine congruency or supplementary relationships.

Area, Perimeter, and Geometric Means

The area of a trapezoid is calculated using $\frac{1}{2}h(b_1+b_2)$, where $h$ is height and $b_n$ are the bases. The area of a kite or rhombus is defined as $\frac{1}{2}d_1 d_2$, where $d$ represents the diagonals. For general polygons, the area is $\frac{1}{2} \times a \times p$, where $a$ is the apothem and $p$ is the perimeter. Perimeter is found by adding all the sides. In a triangle, a midsegment is parallel to the base and equals $\frac{1}{2}(\text{base length})$. For a trapezoid midsegment, the length is $\frac{1}{2}(\text{sum of bases})$, illustrated by the example $\frac{1}{2}(10+20)=15$. Geometric mean relationships in right triangles include the altitude rule: $\text{altitude} = \text{square root of } ab$, where $a$ and $b$ are segments of the hypotenuse.

Similarity, Right Triangles, and Trigonometry

When triangles are similar, proportions are used to find missing lengths 99% of the time, often involving cross-multiplication. Example proportions provided include $\frac{9}{12} = \frac{x}{8}$ resulting in $8x=12$ and $x=1.5$, as well as $\frac{x}{8} = \frac{9}{12}$ resulting in $12x=72$ and $x=6$. Similar triangles features proportional sides and congruent angles. Right triangles follow the Pythagorean theorem: $c^2 = a^2+b^2$. This can be used to classify triangles: if $c^2 = a^2+b^2$ it is a right triangle; if $c^2 > a^2+b^2$ it is an obtuse triangle; and if $c^2 < a^2+b^2$ it is an acute triangle. Special right triangles include the 45-45-90 type, where the hypotenuse is the $\text{leg} \times \text{square root of } 2$, and the 30-60-90 type, where the hypotenuse is twice the short leg ($2x$) and the long leg is the $\text{short leg} \times \text{square root of } 3$. Trigonometric functions are defined as $\text{Sine} = \frac{\text{opposite}}{\text{hypotenuse}}$, $\text{Cosine} = \frac{\text{adjacent}}{\text{hypotenuse}}$, and $\text{Tangent} = \frac{\text{opposite}}{\text{adjacent}}$. If the angle is missing, the inverse function (e.g., $\tan^{-1}$, $\text{cos}^{-1}$, $\text{sin}^{-1}$) is used for the calculation.

Circle Geometry and Equations

The circumference of a circle is $2\text{pi}r$ or $\text{pi}d$, and the area is $\text{pi}r^2$. Arcs and sectors are parts of the circle: arc length is $\frac{\text{angle}}{360} \times 2\text{pi}r$ and sector area is $\frac{\text{angle}}{360} \times \text{pi}r^2$. In circle angle relationships, a central angle equals the intercepted arc, while an inscribed angle (vertex on the circle) equals $\frac{1}{2}$ the intercepted arc (e.g., arc $100 = \text{angle } 50$). If the vertex is inside the circle, the angle is $\frac{1}{2}(\text{sum of intercepted arcs})$. If the vertex is outside, the angle is $\frac{1}{2}(\text{arc } a - \text{arc } b)$. Chords intersecting inside the circle follow the rule $a \times b = c \times d$. Secants meeting outside follow the rule $\text{outer}(\text{whole}) = \text{outer}(\text{whole})$, specifically $b(a+b) = d(d+c)$. Any line tangent to a circle is perpendicular to the radius, allowing for the use of the Pythagorean theorem (e.g., $r^2+12^2=13^2$ yields $r=5$). If a quadrilateral is inscribed in a circle, opposite angles are supplementary ($\text{angle } 1 + \text{angle } 3 = 180^\text{o}$). The equation of a circle is $(x-h)^2+(y-k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius.

Transformations and Solid Geometry

Transformations include reflection, rotation, translation, and dilation. Reflection over the X-axis changes $(x, y)$ to $(x, -y)$, and reflection over the Y-axis changes $(x, y)$ to $(-x, y)$. Rotations follow specific rules: 90 degrees clockwise (CW) or 270 degrees counter-clockwise (CCW) is $(x, y) \rightarrow (y, -x)$; 180 degrees is $(x, y) \rightarrow (-x, -y)$; and 270 degrees CW or 90 degrees CCW is $(x, y) \rightarrow (-y, x)$. Translations shift coordinates by adding or subtracting from $x$ (left/right) or $y$ (up/down). Dilation involves multiplication by a scale factor. In 3D geometry, Prisms and Cylinders have volume $V = Bh$. Pyramids and Cones have volume $V = \frac{1}{3}Bh$. For spheres, Surface Area is $4\text{pi}r^2$ and Volume is $\frac{4}{3}\text{pi}r^3$. Euler's formula states $\text{faces} + \text{vertices} = \text{edges} + 2$. Scale factors follow specific ratios: perimeter is the scale factor ($k$), area is the factor squared ($k^2$), and volume is the factor cubed ($k^3$).

Logic, Proofs, and Geometric Properties

Triangle congruency is proven via SSS, SAS, ASA, AAS, and HL. CPCTC (Corresponding Parts of Congruent Triangles are Congruent) is used after triangles are proven congruent. Key geometric properties include the Transitive Property (if $AB=BC$ and $BC=CD$, then $AB=CD$), the Reflexive Property (anything equals itself, often used for shared sides), the Symmetric Property (if one thing equals another, it can be flipped), and the Substitution Property (replacing equal values). Algebraic properties such as Addition, Subtraction, Multiplication, and Division also apply to geometric equalities when operations are performed on both sides of an equation.