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
On a coordinate map of a new metro line, one station is at (−2, 9) and another is at (−12, 5). A ticket kiosk will be placed at the midpoint of the segment joining the two stations. What are the coordinates of the kiosk?
- A.
  (−7, 7)
- B.
  (−5, −2)
- C.
  (7/2, −7/2)
- D.
  (7, −7)
Answer: A. (−7, 7)
Average each coordinate: ((−2 + (−12))/2, (9 + 5)/2) = (−7, 7). Choice B halves the differences of the coordinates instead of their sums; C averages the x and y of each station with each other; D writes the correct midpoint with its coordinates reversed.
Question 2
A vertex of a triangle is at (2, 5). The triangle is translated 4 units left and 3 units down, and the image is then reflected over the x-axis. Where does that vertex end up?
- A.
  (−2, −8)
- B.
  (−2, −2)
- C.
  (−2, 2)
- D.
  (6, −2)
Answer: B. (−2, −2)
Translate first: (2 − 4, 5 − 3) = (−2, 2). Then reflect over the x-axis: (−2, −2). Choice A reflects before translating, so it gets (2, −5) → (−2, −8); C forgets the reflection; D moves 4 units right instead of left.
Question 3
Which of the following is an equation of the perpendicular bisector of the segment with endpoints (−2, 1) and (4, 5)?
- A.
  y = (2/3)x + 7/3
- B.
  y = (3/2)x + 3/2
- C.
  y = −(3/2)x − 2
- D.
  y = −(3/2)x + 9/2
Answer: D. y = −(3/2)x + 9/2
The midpoint is (1, 3) and the segment's slope is 4/6 = 2/3, so the bisector has slope −3/2: y − 3 = −(3/2)(x − 1), or y = −(3/2)x + 9/2. Choice A keeps the original slope; B takes the reciprocal without the negative; C has the right slope but passes through the endpoint (−2, 1) instead of the midpoint.
Question 4
A triangle has vertices at (0, 0), (4, 0), and (0, 7). It is dilated about the origin by a scale factor of 3. What is the area of the image, in square units?
- A.
  17
- B.
  42
- C.
  126
- D.
  378
Answer: C. 126
The original area is ½ × 4 × 7 = 14. Lengths triple, so area is multiplied by 3² = 9: 14 × 9 = 126. Choice A adds 3 to the area; B multiplies the area by 3, the linear factor; D multiplies by 3³, a volume factor.
Question 5
Three villages are at (0, 0), (8, 0), and (0, 6) on a map measured in kilometers. A water tower is to be built at the one point that is the same distance from all three villages. Where should the tower be built?
- A.
  (8/3, 2)
- B.
  (3, 4)
- C.
  (4, 0)
- D.
  (4, 3)
Answer: D. (4, 3)
The villages form a right triangle, and the point equidistant from all three vertices is the midpoint of the hypotenuse from (8, 0) to (0, 6): (4, 3), which is 5 km from each. Choice A averages all three points (the centroid); B reverses the coordinates; C is the midpoint of the side from (0, 0) to (8, 0), which is 4 km from two villages but 7.2 km from the third.

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