1.3 Geometry & Trigonometry Fundamentals

1.3 Geometry & Trigonometry Fundamentals

Geometry and trigonometry questions on the PMAEE evaluate spatial reasoning, coordinate graphing, and trigonometric relations. Memorizing formulas and recognizing geometric shortcuts are key to saving time.

Angles

An angle is formed by two rays sharing a common endpoint. Angles are classified by their measures:

• Acute angle: Less than $90^\circ$
• Right angle: Exactly $90^\circ$
• Obtuse angle: Greater than $90^\circ$ but less than $180^\circ$
• Straight angle: Exactly $180^\circ$
• Reflex angle: Greater than $180^\circ$ but less than $360^\circ$

Angle Pairs

• Complementary angles: Two angles whose sum is exactly $90^\circ$.
• Supplementary angles: Two angles whose sum is exactly $180^\circ$.
• Vertical angles: Opposing angles formed by intersecting lines; they are always equal.

Parallel Lines cut by a Transversal

When two parallel lines are intersected by a third line, called a transversal, specific angle relationships are established:

• Corresponding angles are equal (e.g., top-right positions).
• Alternate interior angles are equal (inside parallel lines, opposite sides of transversal).
• Alternate exterior angles are equal (outside parallel lines, opposite sides of transversal).
• Consecutive interior angles are supplementary (inside, same side of transversal).

Triangles

A triangle is a three-sided polygon. The sum of the interior angles of any triangle is always $180^\circ$.

Area and Perimeter

• Perimeter: $P = a + b + c$ (sum of all sides).
• Area: $A = \frac{1}{2}bh$, where $b$ is the base and $h$ is the altitude perpendicular to the base.
• Heron's Formula (when height is unknown but all three sides are known):

$A = \sqrt{s(s-a)(s-b)(s-c)}$

where $s$ is the semi-perimeter: $s = \frac{a+b+c}{2}$.

The Pythagorean Theorem

In any right triangle, the square of the length of the hypotenuse ($c$) is equal to the sum of the squares of the lengths of the legs ($a$ and $b$):

$a^2 + b^2 = c^2$

PMAEE frequently uses Pythagorean triples—sets of three integers that satisfy the theorem, saving you calculation time:

• $3-4-5$ (and multiples like $6-8-10, 9-12-15$)
• $5-12-13$ (and multiples like $10-24-26$)
• $8-15-17$
• $7-24-25$

Special Right Triangles

Certain right triangles have fixed angle-to-side relationships:

1. $45^\circ - 45^\circ - 90^\circ$ Right Triangle (Isosceles Right Triangle):
   • The ratio of sides is $1 : 1 : \sqrt{2}$.
   • If the legs are length $x$, the hypotenuse is $x\sqrt{2}$.
2. $30^\circ - 60^\circ - 90^\circ$ Right Triangle:
   • The ratio of sides is $1 : \sqrt{3} : 2$.
   • If the side opposite the $30^\circ$ angle (shortest leg) is $x$, the side opposite the $60^\circ$ angle (longer leg) is $x\sqrt{3}$, and the hypotenuse is $2x$.

Basic Trigonometric Ratios

Trigonometric ratios express the relationships between the acute angles of a right triangle and its sides:

• Sine ($\sin \theta$) = $\frac{\text{Opposite}}{\text{Hypotenuse}}$ (SOH)
• Cosine ($\cos \theta$) = $\frac{\text{Adjacent}}{\text{Hypotenuse}}$ (CAH)
• Tangent ($\tan \theta$) = $\frac{\text{Opposite}}{\text{Adjacent}}$ (TOA)

Reciprocal functions:

• Cosecant ($\csc \theta$) = $\frac{1}{\sin \theta} = \frac{\text{Hypotenuse}}{\text{Opposite}}$
• Secant ($\sec \theta$) = $\frac{1}{\cos \theta} = \frac{\text{Hypotenuse}}{\text{Adjacent}}$
• Cotangent ($\cot \theta$) = $\frac{1}{\tan \theta} = \frac{\text{Adjacent}}{\text{Opposite}}$

Trigonometric Values for Common Angles

Polygons

A polygon is a closed, two-dimensional shape with straight sides. A regular polygon has all sides equal and all angles equal.

Formulas

• Sum of interior angles: $(n - 2) \cdot 180^\circ$, where $n$ is the number of sides.
• Individual interior angle of a regular polygon: $\frac{(n-2) \cdot 180^\circ}{n}$
• Sum of exterior angles: Always $360^\circ$ for any convex polygon.
• Individual exterior angle of a regular polygon: $\frac{360^\circ}{n}$

Quadrilaterals

Quadrilaterals are four-sided polygons. Key classes include:

• Parallelogram: Opposite sides parallel and equal. Diagonals bisect each other.
• Rectangle: Parallelogram with four right angles. Diagonals are equal.
• Rhombus: Parallelogram with four equal sides. Diagonals are perpendicular.
• Square: Parallelogram with four equal sides and four right angles.
• Trapezoid: Quadrilateral with exactly one pair of parallel sides (called bases). Area $A = \frac{a+b}{2} \cdot h$.

Circles

A circle is the set of all points in a plane at a given distance from a center.

• Diameter: $d = 2r$, where $r$ is the radius.
• Circumference: $C = 2\pi r = \pi d$
• Area: $A = \pi r^2$
• Arc Length: $L = \frac{\theta}{360^\circ} \cdot 2\pi r$, where $\theta$ is the central angle in degrees.
• Sector Area: $A_s = \frac{\theta}{360^\circ} \cdot \pi r^2$
• Chord: A line segment connecting two points on a circle. The longest chord is the diameter.

Coordinate Geometry

Coordinate geometry maps geometric shapes onto the Cartesian plane using ordered pairs $(x, y)$.

Key Formulas

Given two points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$:

• Slope ($m$): Measures the steepness of a line. $m = \frac{y_2 - y_1}{x_2 - x_1}$
• Distance Formula: Derived from the Pythagorean theorem. $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
• Midpoint Formula: Finds the center point between two coordinates. $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

Slopes of Parallel and Perpendicular Lines

• Parallel lines: Have identical slopes ($m_1 = m_2$).
• Perpendicular lines: Have slopes that are negative reciprocals ($m_1 \cdot m_2 = -1$). For example, if a line has a slope of $3/4$, the line perpendicular to it has a slope of $-4/3$.

graph TD
    Quad["Quadrilaterals (4 sides)"] --> Parallelogram["Parallelogram (Opposite sides parallel)"]
    Quad --> Trapezoid["Trapezoid (1 pair parallel sides)"]
    Parallelogram --> Rectangle["Rectangle (4 right angles)"]
    Parallelogram --> Rhombus["Rhombus (4 equal sides)"]
    Rectangle --> Square["Square (4 equal sides + 4 right angles)"]
    Rhombus --> Square