How to Solve ACT Math: Coordinate Geometry (Slope, Distance, Midpoint, Equation of a Circle)

How to Solve ACT Math: Coordinate Geometry (Slope, Distance, Midpoint, Equation of a Circle)

Introduction

"Mastering coordinate geometry on the ACT Math section can boost your score by 3–5 points—enough to turn a ‘good’ score into a ‘great’ one. These formulas help you find slopes of hills, distances between cities, and even the center of a circular race track—all in under a minute per question."

What You Need To Know First

1. Plotting points on a coordinate plane (x, y).
2. Basic algebra (solving for variables, plugging in numbers).
3. Square roots and exponents (for distance formula).

Key Vocabulary

Formulas To Know

1. Slope Formula

Formula: [ m = \frac{y_2 - y_1}{x_2 - x_1} ] - ( m ) = slope - ( (x_1, y_1) ) = first point - ( (x_2, y_2) ) = second point Memorise This.

2. Distance Formula

Formula: [ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ] - ( d ) = distance between two points - ( (x_1, y_1) ) and ( (x_2, y_2) ) = two points Memorise This.

3. Midpoint Formula

Formula: [ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) ] - ( (x_1, y_1) ) and ( (x_2, y_2) ) = two points Memorise This.

4. Equation of a Circle

Formula: [ (x - h)^2 + (y - k)^2 = r^2 ] - ( (h, k) ) = center of the circle - ( r ) = radius Memorise This.

5. Parallel & Perpendicular Slopes

• Parallel lines: Same slope.
• Perpendicular lines: Slopes are negative reciprocals (e.g., ( m_1 = 2 ), ( m_2 = -\frac{1}{2} )). Memorise This.

Step-by-Step Method

How to Solve Any Coordinate Geometry Problem on the ACT

1. Read the question. Underline what’s being asked (slope, distance, midpoint, circle equation).
2. Identify the points or equation. Write down the given coordinates or circle equation.
3. Choose the right formula. Match the question to the formula (slope, distance, midpoint, or circle).
4. Plug in the numbers. Substitute the values into the formula.
5. Simplify step-by-step. Do one operation at a time to avoid mistakes.
6. Check units and signs. Did you mix up ( x ) and ( y )? Did you forget a negative?
7. Answer the question. Make sure your final answer matches what’s asked (e.g., "Find the radius" vs. "Find the center").

Worked Example (Using the Steps)

Question: What is the distance between the points ( (3, -2) ) and ( (7, 1) )?

Step 1: Underline → "distance between the points." Step 2: Points given: ( (3, -2) ) and ( (7, 1) ). Step 3: Formula needed: Distance formula. Step 4: Plug in: [ d = \sqrt{(7 - 3)^2 + (1 - (-2))^2} ] Step 5: Simplify: [ d = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 ] Step 6: Check signs → ( 1 - (-2) = 3 ) (correct). Step 7: Answer: 5.

Worked Examples

Example 1 – Basic (Slope)

Question: What is the slope of the line passing through ( (1, 4) ) and ( (5, 12) )?

Solution:
1. Points: ( (1, 4) ) and ( (5, 12) ).
2. Slope formula: ( m = \frac{y_2 - y_1}{x_2 - x_1} ).
3. Plug in: ( m = \frac{12 - 4}{5 - 1} = \frac{8}{4} = 2 ). Answer: 2

What we did and why: We used the slope formula directly. No tricks—just plug and chug.

Example 2 – Medium (Midpoint + Distance)

Question: Point ( A ) is at ( (2, 3) ) and point ( B ) is at ( (8, -1) ). What is the midpoint of ( AB ), and what is the distance between ( A ) and ( B )?

Solution: Part 1: Midpoint
1. Midpoint formula: ( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) ).
2. Plug in: ( \left( \frac{2 + 8}{2}, \frac{3 + (-1)}{2} \right) = (5, 1) ).

Part 2: Distance
1. Distance formula: ( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ).
2. Plug in: ( d = \sqrt{(8 - 2)^2 + (-1 - 3)^2} = \sqrt{6^2 + (-4)^2} = \sqrt{36 + 16} = \sqrt{52} = 2\sqrt{13} ).

Answer: Midpoint = (5, 1), Distance = ( 2\sqrt{13} )

What we did and why: We split the question into two parts. First, we found the midpoint by averaging the ( x ) and ( y ) values. Then, we used the distance formula, remembering to square the differences and simplify the square root.

Example 3 – Exam-Style (Circle Equation)

Question: A circle has the equation ( (x + 1)^2 + (y - 4)^2 = 36 ). What is the area of the circle?

Solution:
1. Circle equation: ( (x - h)^2 + (y - k)^2 = r^2 ).
2. Compare to given: ( (x + 1)^2 + (y - 4)^2 = 36 ). - ( h = -1 ), ( k = 4 ), ( r^2 = 36 ).
3. Find radius: ( r = \sqrt{36} = 6 ).
4. Area formula: ( A = \pi r^2 ).
5. Plug in: ( A = \pi (6)^2 = 36\pi ).

Answer: ( 36\pi )

What we did and why: The question disguised the radius by giving ( r^2 ). We had to take the square root first, then use the area formula. Always check if the equation is in standard form!

Common Mistakes

Exam Traps

1-Minute Recap

"Listen up—this is your last-minute cheat sheet for ACT coordinate geometry. Memorize these four formulas: slope, distance, midpoint, and circle equation. For slope, it’s rise over run—subtract ( y )’s over ( x )’s. Distance? Square the differences, add them, take the square root. Midpoint? Average the ( x )’s and ( y )’s. Circle equation? ( (x - h)^2 + (y - k)^2 = r^2 )—center is ( (h, k) ), radius is ( \sqrt{r^2} ). Watch for traps: if they ask for radius but give ( r^2 ), take the square root. If lines are perpendicular, slopes are negative reciprocals. Underline the question, pick the right formula, plug in carefully, and double-check signs. You’ve got this!