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ACT Coordinate geometry practice questions

Coordinate geometry sits in the Geometry reporting category on the ACT Math section. Prep36 has 14 questions tagged to this skill, every one with a worked explanation. Three of them are below in full, with the reasoning, so you can see what the skill actually looks like before you practise it.

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What Coordinate geometry covers on the ACT

Within this skill the Prep36 bank covers distance formula, equation of a circle, midpoint formula, perimeter of a right triangle on the plane, finding a missing endpoint from a midpoint and slope from two points.

Math is 45 questions in 50 minutes, which is about 66 seconds each, and it is one of the three sections that average into your composite. The difficulty split across the 14 questions tagged here is 4 easier, 7 medium and 3 harder.

Questions are original, written to the Enhanced ACT format. Prep36 is not affiliated with ACT, Inc.

3 coordinate geometry questions, with the reasoning

Question 1 easy

In the standard (x, y) coordinate plane, what is the midpoint of the line segment with endpoints (-3, 7) and (5, -1)?

  1. A. (1, 3) correct
  2. B. (4, -4)
  3. C. (2, 6)
  4. D. (1, 4)
Why: The midpoint is the average of the endpoints coordinate by coordinate: x = (-3 + 5)/2 = 1 and y = (7 + (-1))/2 = 3, so the midpoint is (1, 3), choice A. The choice (2, 6) adds the coordinates but never divides by 2. The choice (4, -4) is a difference rather than an average. The choice (1, 4) gets x right but mishandles the negative in the y average. Tip: a midpoint must land between the two endpoints. Since the y values are 7 and -1, any answer with y outside that range can be eliminated on sight.
Question 2 medium

In the standard (x,y) coordinate plane, points A(−3, 5) and B(7, −1) are the endpoints of a diameter of a circle. Which of the following is an equation of that circle?

  1. A. (x − 2)² + (y − 2)² = 34 correct
  2. B. (x − 2)² + (y − 2)² = 136
  3. C. (x + 2)² + (y + 2)² = 34
  4. D. (x − 2)² + (y − 2)² = √34
Why: The center is the midpoint of AB: ((−3 + 7)/2, (5 + (−1))/2) = (2, 2), which forces the form (x − 2)² + (y − 2)². The diameter is √(10² + 6²) = √136, so r = √136/2 and r² = 136/4 = 34, giving (x − 2)² + (y − 2)² = 34. Choice 136 uses the squared diameter instead of the squared radius, the √34 version forgets that the right side is r² and not r, and the (x + 2)² version flips the sign of a positive center. Tip: In (x − h)² + (y − k)² = r², the signs inside the parentheses are the opposite of the center coordinates and the right side is always r squared, so check both before choosing.
Question 3 hard

In the standard (x, y) coordinate plane, a circle has the equation x² + y² - 6x + 4y - 12 = 0. What is the radius of the circle?

  1. A. 2√3
  2. B. √13
  3. C. 5 correct
  4. D. 25
Why: Complete the square in both variables. Half of -6 squared is 9 and half of 4 squared is 4, so (x² - 6x + 9) + (y² + 4y + 4) = 12 + 9 + 4, which gives (x - 3)² + (y + 2)² = 25. Since the right side is r², the radius is √25 = 5. The distractors stop one step early or skip the completion: 25 is r², √13 mishandles the added constants, and 2√3 is just √12 from the original constant. Tip: whatever you add inside the parentheses must be added to the right side too, and the final number is always r squared, never the radius itself.

The explanation is the point. Getting a question wrong tells you nothing you did not already suspect; knowing which step you missed is what changes the next one.

Practise the rest of the 14

The other 11 questions on this skill are in the free question bank, filterable by difficulty, each with the same worked explanation. No account and no card.

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Frequently asked questions

How many ACT coordinate geometry questions does Prep36 have?
14, each tagged to this skill and each with a worked explanation.
Are these real ACT questions?
No. They are original questions written to the Enhanced ACT format. Prep36 is not affiliated with ACT, Inc.
Is the Math section part of the composite?
Yes. On the Enhanced ACT the composite is the average of English, Math and Reading. Science is optional and reported separately.
Keep readingACT Math practiceEvery Math skill in one place.How many questions can you miss?The miss budget for every score.ACT Score CalculatorTurn raw scores into a composite.
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