ACT · Math · Preparing for Higher Math: Geometry

Coordinate geometry

Coordinate geometry is part of Preparing for Higher Math: Geometry in the ACT Math section. The questions below are written in the style of the real test, each with the answer and a worked explanation.

Question types you will see here include midpoint of a segment, distance between two points, translation, reflection, rotation and area or perimeter from coordinates.

Common traps: adding the coordinates without dividing by 2; subtracting the midpoint instead of doubling; averaging x with y.

Sample questions

Question 1
Point M(3, −2) is the midpoint of segment AB, and A is (−1, 4). What is the sum of the coordinates of B?
- A.
  −2
- B.
  −1
- C.
  1
- D.
  4
Answer: B. −1
B = 2M − A = (6 + 1, −4 − 4) = (7, −8), so the sum is 7 + (−8) = −1. Choice A (−2) finds M − A instead of 2M − A. Choice C (1) gives the sum of M's coordinates. Choice D (4) adds the coordinates of A and M instead of doubling M and subtracting A.
Question 2
In a drawing app, a point at (−3, 6) is rotated 90° clockwise about the origin. Where does the point end up?
- A.
  (−6, −3)
- B.
  (3, −6)
- C.
  (6, −3)
- D.
  (6, 3)
Answer: D. (6, 3)
A 90° clockwise rotation about the origin sends (x, y) to (y, −x), so (−3, 6) → (6, 3). Choice A rotates counterclockwise, (−y, x); B rotates 180°, changing both signs; C only swaps the coordinates, which is a reflection over the line y = x.
Question 3
A straight footpath in a park plan follows the perpendicular bisector of the segment joining two lamp posts at (1, 7) and (7, 3). What is the y-intercept of the footpath's line?
- A.
  −1
- B.
  11/2
- C.
  23/3
- D.
  11
Answer: A. −1
The midpoint is (4, 5) and the segment's slope is −4/6 = −2/3, so the path's slope is 3/2. Then 5 = (3/2)(4) + b gives b = −1. Choice B uses slope 3/2 through the endpoint (1, 7); C keeps the original slope −2/3; D uses −3/2, the reciprocal with the wrong sign.
Question 4
In which of the following sets do all three points lie on a single line?
- A.
  (0, 1), (3, 7), (5, 12)
- B.
  (1, 0), (3, 4), (6, 9)
- C.
  (−1, −1), (2, 5), (4, 10)
- D.
  (0, 1), (2, 5), (5, 11)
Answer: D. (0, 1), (2, 5), (5, 11)
In the correct set both slopes match: 4/2 = 2 and 6/3 = 2. Every set has slope 2 between its first two points, which is the trap. In choice A the second slope is 5/2; in B it is 5/3; in C it is 5/2. Check both pairs, not just one.
Question 5
A mural plan for a school wall is drawn on a coordinate grid in meters. A rope runs from the corner P(9, 2) through the midpoint M(5, −1) to the opposite corner Q. What are the coordinates of Q?
- A.
  (1, −4)
- B.
  (4, 3)
- C.
  (7, 0.5)
- D.
  (14, 1)
Answer: A. (1, −4)
M is the midpoint of PQ, so Q = 2M − P = (10 − 9, −2 − 2) = (1, −4). Choice B ((4, 3)) subtracts P from M the wrong way round, P − M. Choice C ((7, 0.5)) is the midpoint of P and M, not the endpoint. Choice D ((14, 1)) adds the coordinates of P and M without doubling M.

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