ACT Math: Preparing for Higher Math: Geometry

ACT Math Overview

The ACT Math section gives you 60 minutes to answer 60 questions, so you're working at a pace of about one question per minute. The section covers algebra, geometry, trigonometry, basic arithmetic, and data analysis, all in multiple-choice format with five answer choices per question.

Here's a breakdown of the question types you'll encounter:

• Pre-algebra and elementary algebra: Basic operations, fractions, decimals, percentages, exponents, and simple linear equations.
• Intermediate algebra and coordinate geometry: Quadratic equations, factoring, inequalities, systems of equations, and graphing in the coordinate plane.
• Plane geometry: Angles, lines, triangles, quadrilaterals, circles, and other plane figures. Expect area, perimeter, and volume calculations.
• Trigonometry: Basic trig concepts, ratios (sine, cosine, and tangent), and solving trigonometric equations.
• Data analysis and statistics: Interpreting graphs, charts, and tables; calculating measures of central tendency, probability, and basic statistics.

This guide focuses specifically on the geometry concepts tested in the ACT Math section.

Tips and Tricks

These strategies won't replace solid preparation, but they can help you pick up extra points and manage your time more effectively.

1. Learn the formulas and understand the theory first. Then practice with timed sample tests to build speed. When you actually understand why a formula works, applying it under pressure gets much easier.

2. Use process of elimination. The multiple-choice format works in your favor. Even if you're unsure, crossing out answers you know are wrong improves your odds significantly.

3. Plug in the answer choices. When a question asks you to solve for a variable and you're stuck, try substituting each answer choice back into the equation. This is especially useful on algebra-heavy problems and can save you real time.

4. Stay calm and pace yourself. If a question is taking too long, mark it and move on. You can always come back. Getting flustered on one problem can cause careless mistakes on the next three.

Geometry

Geometry doesn't have its own separate section on the ACT. Instead, it shows up woven into questions throughout the test. Here are the core geometric concepts you should know:

• Triangles: Know the properties of different triangle types (equilateral, isosceles, scalene, right). The angles in any triangle always add up to $180°$. The Pythagorean theorem ($a^2 + b^2 = c^2$) applies to right triangles and comes up constantly.
• Circles: Understand the relationships between the radius, diameter, chords, and arcs. Know central angles (vertex at the center) and inscribed angles (vertex on the circle, which are half the central angle intercepting the same arc). Memorize the formulas: circumference = $2\pi r$ and area = $\pi r^2$.
• Quadrilaterals: Learn the properties of rectangles, squares, parallelograms, and trapezoids. The ACT frequently asks you to find a missing angle or side length using these properties.
• Polygons: For any polygon with $n$ sides, the sum of interior angles is $(n-2) \times 180°$. Each exterior angle of a regular polygon is $\frac{360°}{n}$.
• Coordinate geometry: Practice working with points, lines, and equations on the coordinate plane. You should be comfortable finding the distance between two points ($\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$), the slope of a line ($\frac{y_2 - y_1}{x_2 - x_1}$), the midpoint ($\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$), and where two lines intersect. These questions trip up a lot of students, so practice them in different scenarios.
• Three-dimensional figures: Not as heavily tested as 2D geometry, but you should know the volume and surface area formulas for common solids. For example: volume of a cylinder = $\pi r^2 h$, volume of a sphere = $\frac{4}{3}\pi r^3$, volume of a rectangular prism = $l \times w \times h$.
• Transformations: Know how reflections, rotations, translations, and dilations affect figures on the coordinate plane. The questions tend to be straightforward if you've memorized the rules (for example, reflecting a point $(x, y)$ over the x-axis gives $(x, -y)$).
• Memorize the basic formulas for area, perimeter, and diagonals of common shapes. The ACT does not provide a formula sheet, so you need these committed to memory.

Applying Your Knowledge: Practice Problems

1. In triangle ABC, angle A measures 40 degrees, and angle B measures 75 degrees. What is the measure of angle C?

EXPLANATION

The sum of angles in a triangle is always $180°$.

1. Substitute the known values: $40 + 75 + \angle C = 180$
2. Combine: $115 + \angle C = 180$
3. Solve: $\angle C = 180 - 115 = 65°$

Angle C = 65 degrees.

2. A rectangle has a length of 12 units and a diagonal of 13 units. What is the width of the rectangle?

EXPLANATION

A rectangle's diagonal splits it into two right triangles. The length and width are the two legs, and the diagonal is the hypotenuse.

Let the width = $w$.

1. Set up the equation: $12^2 + w^2 = 13^2$

2. Simplify: $144 + w^2 = 169$

3. Isolate $w^2$: $w^2 = 169 - 144 = 25$

4. Solve: $w = \sqrt{25} = 5$

The width is 5 units.

3. Triangle XYZ has side lengths XY = 8 units, XZ = 10 units, and YZ = 12 units. Is triangle XYZ a right triangle?

EXPLANATION

To check if a triangle is a right triangle, test whether the Pythagorean theorem holds. The longest side must be the hypotenuse, so $YZ = 12$ goes on the right side of the equation.

1. Check if $XY^2 + XZ^2 = YZ^2$
2. Left side: $8^2 + 10^2 = 64 + 100 = 164$
3. Right side: $12^2 = 144$
4. Compare: $164 \neq 144$

Since the two sides aren't equal, triangle XYZ is not a right triangle. (Notice that $164 > 144$, which actually tells you the triangle is acute, since the sum of the squares of the two shorter sides is greater than the square of the longest side.)

Conclusion

Geometry on the ACT is very formula-driven. If you memorize the key formulas, understand the properties of basic shapes, and practice applying them under timed conditions, you'll be well-prepared. Focus especially on triangles, circles, and coordinate geometry, since those appear most frequently. For additional practice, work through full-length timed sections so you get comfortable with the pacing.