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ACT Math Geometry and Trigonometry Question Types cheat sheet - grade 11-12

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Math Grade 11-12

ACT Math Geometry and Trigonometry Question Types Cheat Sheet

A printable reference covering plane geometry, coordinate geometry, right triangle trigonometry, area formulas, slope, distance, and ACT question strategies for grades 11-12.

TopicsShapesTrigonometryArea and PerimeterGraphsAlgebraGeometryStudy Guide
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This cheat sheet covers the geometry and trigonometry question types most often tested on the ACT Math section. Students need these formulas because ACT problems usually reward quick recognition, setup, and substitution. The sheet connects plane geometry, coordinate geometry, and trigonometry so students can choose an efficient method under time pressure.

The most important ideas include area, perimeter, triangle relationships, circle formulas, slope, distance, midpoint, and right triangle trig ratios. Many ACT problems combine diagrams with algebra, so labeling given values clearly is essential. For trigonometry, students should know sin⁡θ\sin \theta, cos⁡θ\cos \theta, tan⁡θ\tan \theta, special right triangles, and when to use the Pythagorean theorem.

Key Facts

  • The area of a triangle is A=12bhA = \frac{1}{2}bh, where bb is the base and hh is the perpendicular height.
  • The Pythagorean theorem is a2+b2=c2a^2 + b^2 = c^2, where cc is the hypotenuse of a right triangle.
  • The circumference of a circle is C=2πrC = 2\pi r and the area is A=πr2A = \pi r^2.
  • The slope of a line through (x1,y1)(x_1,y_1) and (x2,y2)(x_2,y_2) is m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}.
  • The distance between (x1,y1)(x_1,y_1) and (x2,y2)(x_2,y_2) is d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.
  • The midpoint of a segment is M=(x1+x22,y1+y22)M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right).
  • For a right triangle, sin⁡θ=oppositehypotenuse\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}, cos⁡θ=adjacenthypotenuse\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}, and tan⁡θ=oppositeadjacent\tan \theta = \frac{\text{opposite}}{\text{adjacent}}.
  • In a 45∘−45∘−90∘45^\circ-45^\circ-90^\circ triangle, the side ratio is 1:1:21:1:\sqrt{2}, and in a 30∘−60∘−90∘30^\circ-60^\circ-90^\circ triangle, the side ratio is 1:3:21:\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=riserunm = \frac{\text{rise}}{\text{run}}.
Trigonometric Ratio
A trigonometric ratio compares two sides of a right triangle using an angle, such as sin⁡θ\sin \theta, cos⁡θ\cos \theta, or tan⁡θ\tan \theta.
Special Right Triangle
A special right triangle has side ratios that can be memorized, such as 45∘−45∘−90∘45^\circ-45^\circ-90^\circ and 30∘−60∘−90∘30^\circ-60^\circ-90^\circ triangles.

Common Mistakes to Avoid

  • Using the slanted side as height in an area formula is wrong because hh in A=12bhA = \frac{1}{2}bh or A=bhA = bh must be perpendicular to the base.
  • Confusing radius and diameter is wrong because circle formulas use rr, and the diameter is d=2rd = 2r.
  • Subtracting coordinates in the wrong order for slope can give an incorrect sign because m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1} must use the same point order in the numerator and denominator.
  • Using sin⁡θ\sin \theta, cos⁡θ\cos \theta, or tan⁡θ\tan \theta with the wrong reference angle is wrong because opposite and adjacent sides change depending on θ\theta.
  • Assuming a diagram is drawn to scale is risky because ACT geometry figures may not be exact unless lengths, angles, or relationships are stated.

Practice Questions

  1. 1 A right triangle has legs of length 66 and 88. What is the length of the hypotenuse?
  2. 2 Find the slope and midpoint of the segment with endpoints (2,5)(2,5) and (10,1)(10,1).
  3. 3 A circle has radius 77. Find its circumference and area in terms of π\pi.
  4. 4 An ACT problem gives a triangle diagram with no right angle mark but one side looks vertical and another looks horizontal. Explain why you should not automatically use the Pythagorean theorem.

Understanding ACT Math Geometry and Trigonometry Question Types

ACT geometry questions often test whether a picture is trustworthy. A diagram may not be drawn to scale, even when it looks obvious. A line that appears horizontal may not be horizontal.

Two segments that look equal are not equal unless matching tick marks, labels, or a stated relationship proves it. Treat every diagram as a record of given facts, not as a photograph.

Mark angle measures, parallel lines, right angles, and equal lengths as soon as you see them. This reduces missed connections later in the problem.

Many geometry problems become shorter when you search for structure before calculating. Parallel lines create equal angle relationships when a transversal crosses them. Vertical angles are equal.

The interior angles of any triangle total one hundred eighty degrees. Similar triangles have matching angles and proportional side lengths, which makes them especially useful when a small shape sits inside a larger one. Congruent figures have exactly the same size and shape, so corresponding measurements match directly.

On the ACT, an unknown length may come from a proportion rather than a long chain of separate calculations. Keep corresponding vertices in the same order when writing a proportion.

Coordinate geometry connects visual shapes to algebra. A slope describes rate of change, so it can show whether lines are parallel or perpendicular. Parallel nonvertical lines share a slope.

Perpendicular nonvertical lines have slopes whose product is negative one. This helps with questions about rectangles, squares, and right angles placed on a grid. A horizontal line has zero slope, while a vertical line has undefined slope.

Students often make sign errors when subtracting coordinates. Writing each subtraction with parentheses prevents this. Coordinate questions can describe real tasks such as finding the steepness of a wheelchair ramp, locating the center of a map route, or checking the shortest straight path between two points.

Trigonometry is most useful when a right triangle links an angle to a missing side. First identify the hypotenuse. It is opposite the right angle and is always the longest side.

Then name opposite and adjacent relative to the marked angle, since those names change if the chosen angle changes. Calculator settings matter. ACT angle questions normally use degrees, so a calculator in radian mode can produce a wrong result from correct work.

Exact special triangle values can save time and avoid rounding. When an answer choice is an estimate, check whether your result has a sensible size. A sine or cosine value for an acute angle must lie between zero and one.

Area answers use square units, while perimeter and distance use ordinary units. These small checks catch errors before they cost a point.