Coordinate Geometry and Trigonometry — ACT Math

Introduction

Coordinate geometry bridges algebra and geometry — and the ACT tests both the distance formula AND basic trig, making this section a two-for-one opportunity.

Coordinate geometry and trigonometry together account for 8-12 questions per ACT. Trig questions (sin, cos, tan, SOHCAHTOA) appear in the medium-hard range and are frequently skipped by unprepared students.

By the end of this lesson you will be able to:

Apply the distance formula, midpoint formula, and slope relationships
Write and interpret equations of circles and lines in the coordinate plane
Evaluate sine, cosine, and tangent using SOHCAHTOA and the unit circle

You'll identify the center and radius of a circle from its general equation by completing the square — combining algebra and coordinate geometry.

The Concept

The Core Rule

Distance: d = √((x₂−x₁)² + (y₂−y₁)²). Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2). Circle standard form: (x−h)² + (y−k)² = r² with center (h,k) and radius r. SOHCAHTOA: sinθ = Opposite/Hypotenuse, cosθ = Adjacent/Hypotenuse, tanθ = Opposite/Adjacent.

How the ACT tests this

Asking for the distance or midpoint between two given points
Giving a circle equation and asking for center, radius, or whether a point lies on it
Using a right triangle diagram with a labeled angle and asking for a trig ratio

Coordinate Plane Formulas

Distance formula is the Pythagorean theorem: d² = (Δx)² + (Δy)². Midpoint is the average of the coordinates. Slope = rise/run = (y₂−y₁)/(x₂−x₁). Perpendicular slopes: m₂ = −1/m₁.

Horizontal lines have slope 0; vertical lines have undefined slope
Parallel lines: same slope, different y-intercepts
y = mx + b is slope-intercept; Ax + By = C is standard form

Circles in the Coordinate Plane

Standard form: (x−h)² + (y−k)² = r². To convert from general form, complete the square on x and y separately. A point (a,b) is inside, on, or outside the circle if (a−h)² + (b−k)² is less than, equal to, or greater than r².

Completing the square: x² + bx → (x + b/2)² − (b/2)²
Center is (h, k), NOT (−h, −k) — watch the signs in the equation
Diameter endpoints: center is their midpoint; radius = half the distance between them

Trigonometry

SOHCAHTOA applies in right triangles. Key values to memorize: sin30°=1/2, cos30°=√3/2, tan30°=1/√3; sin45°=cos45°=√2/2, tan45°=1; sin60°=√3/2, cos60°=1/2, tan60°=√3. Pythagorean identity: sin²θ + cos²θ = 1.

Complementary angles: sinθ = cos(90°−θ)
Law of Sines: a/sinA = b/sinB = c/sinC (any triangle)
Law of Cosines: c² = a² + b² − 2ab·cosC (any triangle)

Your strategy

Label the opposite, adjacent, and hypotenuse sides relative to the given angle before applying SOHCAHTOA
For circle center-radius: rewrite in standard form first; signs inside parentheses flip
Use the distance formula as written — don't try to do it mentally
For trig problems, draw a right triangle with legs labeled if no diagram is given

Worked Examples

Easy Example 1 Adding The Horizontal And Vertical Distances Instead Of Using The Pythagorean Theorem

What is the distance between points (1, 2) and (4, 6)?

A. 3
B. 4
C. 5 (Correct answer)
D. 6
E. 7

Step 1

Apply distance formula: d = √((4−1)² + (6−2)²)

Step 2

d = √(9 + 16) = √25 = 5

Step 3

Distance = 5

Correct answer: C

Why C is correct

Correct: √(9+16) = 5

Why other options are wrong

A: Only computed |4−1| = 3; forgot the y-component

B: Only computed |6−2| = 4; forgot the x-component

D: Added differences without squaring: 3 + 4 = 7, then made arithmetic error

E: Added the horizontal and vertical distances: 3 + 4 = 7

⚠ Trap: Adding the horizontal and vertical distances instead of using the Pythagorean theorem

Medium Example 2 Confusing Sin (Opposite/Hypotenuse) And Cos (Adjacent/Hypotenuse)

In a right triangle, the angle θ is at vertex A. The side opposite θ has length 5, and the hypotenuse has length 13. What is cosθ?

A. 5/13
B. 5/12
C. 12/13 (Correct answer)
D. 13/12
E. 12/5

Step 1

Find the adjacent side: a² + 5² = 13² → a² = 169 − 25 = 144 → a = 12

Step 2

cosθ = Adjacent/Hypotenuse = 12/13

Step 3

Answer: 12/13

Correct answer: C

Why C is correct

Correct: cosθ = 12/13

Why other options are wrong

A: This is sinθ = Opposite/Hypotenuse = 5/13

B: This is sinθ/cosθ = tan-related; uses wrong sides

D: Flipped: Hypotenuse/Adjacent = secθ, not cosθ

E: Adjacent/Opposite = cot-related ratio

⚠ Trap: Confusing sin (Opposite/Hypotenuse) and cos (Adjacent/Hypotenuse)

Hard Example 3 Reading The Center As (6, −4) Directly From The Coefficients Without Completing The Square

What is the center of the circle defined by x² + y² − 6x + 4y − 3 = 0?

A. (3, −2) (Correct answer)
B. (−3, 2)
C. (6, −4)
D. (3, 2)
E. (−3, −2)

Step 1

Group and complete the square: (x² − 6x) + (y² + 4y) = 3

Step 2

(x−3)² − 9 + (y+2)² − 4 = 3 → (x−3)² + (y+2)² = 16

Step 3

Center is (3, −2), radius = 4

Correct answer: A

Why A is correct

Correct: center (3, −2)

Why other options are wrong

B: Sign error: negated both coordinates

C: Read off raw coefficients −6 and +4 directly as center coordinates

D: Got x-coordinate right but missed sign on y: should be −2 not +2

E: Negated both halved coefficients incorrectly

⚠ Trap: Reading the center as (6, −4) directly from the coefficients without completing the square

Strategy Tips

Write SOHCAHTOA on your scratch paper at the start of every trig problem
For circle equations, complete the square on BOTH variables; don't stop at one
Memorize trig values for 30°, 45°, 60° — they appear on every ACT
Midpoint is just the average: add the x-coordinates and divide by 2, same for y
Check that your circle is in (x−h)² + (y−k)² = r² form before reading the center

Common pitfalls

Reading circle center as the raw coefficients without the sign flip

Swapping sin and cos in SOHCAHTOA

Forgetting to complete the square on the y-terms as well as the x-terms

Distance and midpoint: 30 seconds. SOHCAHTOA problems: 60 seconds. Completing the square for a circle: 90 seconds. Always write out the distance formula — mental squaring leads to errors.

Summary

Distance formula = Pythagorean theorem with coordinate differences as legs
Circle center signs flip: (x−h)² + (y−k)² = r² has center (h, k), not (−h, −k)
SOHCAHTOA: always label O, A, H relative to the specific angle you are given

Find the center and radius of x² + y² + 8x − 2y − 8 = 0 by completing the square. Then verify your center by substituting it into the original equation.

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