ACT Math Geometry and Trigonometry Question Types Cheat Sheet

Key Facts

• The area of a triangle is $A = \frac{1}{2}bh$, where $b$ is the base and $h$ is the perpendicular height.
• The Pythagorean theorem is $a^2 + b^2 = c^2$, where $c$ is the hypotenuse of a right triangle.
• The circumference of a circle is $C = 2\pi r$ and the area is $A = \pi r^2$.
• The slope of a line through $(x_1,y_1)$ and $(x_2,y_2)$ is $m = \frac{y_2 - y_1}{x_2 - x_1}$.
• The distance between $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
• The midpoint of a segment is $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$.
• For a right triangle, $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$, $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$, and $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$.
• In a $45^\circ-45^\circ-90^\circ$ triangle, the side ratio is $1:1:\sqrt{2}$, and in a $30^\circ-60^\circ-90^\circ$ triangle, the side ratio is $1:\sqrt{3}:2$.

Vocabulary

Plane Geometry

Plane geometry studies flat shapes such as triangles, circles, quadrilaterals, angles, area, and perimeter.

Coordinate Geometry

Coordinate geometry uses points, lines, slopes, distances, and equations on the coordinate plane.

Hypotenuse

The hypotenuse is the longest side of a right triangle and is always opposite the right angle.

Slope

Slope measures the steepness of a line and is calculated by $m = \frac{\text{rise}}{\text{run}}$.

Trigonometric Ratio

A trigonometric ratio compares two sides of a right triangle using an angle, such as $\sin \theta$, $\cos \theta$, or $\tan \theta$.

Special Right Triangle

A special right triangle has side ratios that can be memorized, such as $45^\circ-45^\circ-90^\circ$ and $30^\circ-60^\circ-90^\circ$ triangles.