Comprehensive Study Guide for Geometry and Trigonometry

Volume of Three-Dimensional Solids

• General Principle of Volume     - Volume represents the amount of space occupied by a three-dimensional object, measured in cubic units (e.g., $cm^3$, $m^3$, $in^3$).     - For uniform solids, the volume ($V$) is generally calculated as the area of the base ($B$) multiplied by the height ($h$): $V = B \times h$.

• Prisms and Cylinders     - Rectangular Prism: A solid where the base is a rectangle. The volume is calculated as length ($l$) times width ($w$) times height ($h$).         - Formula: $V = l \times w \times h$     - Triangular Prism: A solid where the base is a triangle. The volume is the area of the triangular base multiplied by the length of the prism.         - Formula: $V = \frac{1}{2} \times b \times h_{tri} \times L_{prism}$     - Cylinder: A solid with two congruent circular bases. The area of the base is $\pi \times r^2$.         - Formula: $V = \pi \times r^2 \times h$

• Pointed Solids: Pyramids and Cones     - These shapes have one-third the volume of a prism or cylinder with the same base area and height.     - Pyramid: The volume is one-third the product of the base area and the vertical height.         - Formula: $V = \frac{1}{3} \times B \times h$     - Cone: A solid with a circular base tapering to a single point (apex).         - Formula: $V = \frac{1}{3} \times \pi \times r^2 \times h$

• Spheres     - A sphere is a set of all points in space equidistant from a center point. Its volume is derived from the radius ($r$).     - Formula: $V = \frac{4}{3} \times \pi \times r^3$

Right Triangle Trigonometry

• The Pythagorean Theorem     - Applicable only to right-angled triangles. It relates the lengths of the two legs ($a$ and $b$) to the hypotenuse ($c$), which is the side opposite the $90^{\circ}$ angle.     - Formula: $a^2 + b^2 = c^2$     - To find a missing leg: $a = \sqrt{c^2 - b^2}$     - To find the hypotenuse: $c = \sqrt{a^2 + b^2}$

• Trigonometric Ratios (SOH CAH TOA)     - These ratios relate the acute angles of a right triangle to the ratios of its side lengths.     - Sine: $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$     - Cosine: $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$     - Tangent: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$

• Finding Missing Sides     - To solve for a missing side, identify the known angle and one known side, then select the ratio that includes the unknown side. Solve the algebraic equation.     - Example: If $\theta = 30^{\circ}$ and Hypothenuse = $10$, to find the Opposite side ($x$):         - $\sin(30^{\circ}) = \frac{x}{10}$         - $x = 10 \times \sin(30^{\circ})$

• Finding Missing Angles (Inverse Trigonometry)     - If two sides are known, use the inverse trigonometric functions ($\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$) to find the measure of the angle ($\theta$).     - Formula: $\theta = \tan^{-1}\left(\frac{\text{Opposite}}{\text{Adjacent}}\right)$

• Angles of Elevation and Depression     - Angle of Elevation: The angle formed by the line of sight and the horizontal when looking upward.     - Angle of Depression: The angle formed by the line of sight and the horizontal when looking downward.     - Relationship: Because horizontal lines are parallel, the angle of elevation from point A to point B is congruent to the angle of depression from point B to point A (Alternate Interior Angles Theorem).

Circles: Tangents, Chords, and Inscribed Figures

• Tangent Lines     - A line is tangent to a circle if and only if it is perpendicular to the radius at the point of tangency.     - Tangent Segments: Tangent segments from a common external point to a circle are congruent.

• Chords     - A chord is a line segment with both endpoints on the circle.     - Perpendicular Bisector Theorem: If a diameter (or radius) is perpendicular to a chord, then it bisects the chord and its arc.     - Equidistant Chords: In the same circle (or congruent circles), two chords are congruent if and only if they are equidistant from the center.

• Inscribed Angles and Circles     - Inscribed Angle: An angle whose vertex is on the circle and whose sides contain chords. The measure of an inscribed angle is half the measure of its intercepted arc.     - Inscribed Circle (Incircle): A circle contained inside a polygon, tangent to all its sides. The center is the "Incenter," found by the intersection of the angle bisectors of the polygon.

Similarity and Dilation

• Dilations     - A transformation that produces an image that is the same shape as the original, but a different size.     - Scale Factor ($k$): The ratio of any side length in the image to the corresponding side length in the pre-image ($k = \frac{\text{image}}{\text{pre-image}}$).         - If $k > 1$, the dilation is an expansion (enlargement).         - If $0 < k < 1$, the dilation is a contraction (reduction).     - Coordinate Rule: $(x, y) \rightarrow (kx, ky)$, assuming the center of dilation is at the origin $(0,0)$.

• Similarity Transformations     - Two figures are similar ($\sim$) if there is a sequence of rigid motions (translations, reflections, rotations) followed by a dilation that maps one figure onto the other.

• Proving Triangles Similar     - Angle-Angle (AA \sim): If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.     - Side-Side-Side (SSS \sim): If the lengths of the corresponding sides of two triangles are proportional, the triangles are similar.     - Side-Angle-Side (SAS \sim): If an angle of one triangle is congruent to an angle of a second triangle and the sides including these angles are proportional, the triangles are similar.

• Proportions in Triangles     - Triangle Proportionality Theorem (Side-Splitter): If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally.     - Triangle Midsegment Theorem: A midsegment connects the midpoints of two sides; it is parallel to the third side and half its length ($Midsegment = \frac{1}{2} \times \text{Base}$).