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Coordinate geometry — distance, midpoint, slope, line equations — ACT practice questions

Coordinate geometry — distance, midpoint, slope, line equations on the ACT Mathematics test asks students to work in the standard coordinate plane: find slope from two points, take the negative reciprocal for a perpendicular line, and write a line equation from intercepts. Students must also compute distance and midpoint and switch among slope-intercept and standard forms. Items usually present points or intercepts and ask for a slope or an equation. Answer choices look like similar fractions or rearranged equations. Traps include treating a perpendicular slope as the same slope, reversing the slope formula, and swapping intercepts.

  • Section: Mathematics 45 questions on the paper
  • 62 practice questions in the app

Sample questions

Question 1

In the standard (x,y)(x,y) coordinate plane, line kk is perpendicular to a line whose slope is 23\dfrac{2}{3}. What is the slope of line kk?

  1. A−32-\dfrac{3}{2}
  2. B−23-\dfrac{2}{3}
  3. C23\dfrac{2}{3}
  4. D32\dfrac{3}{2}
Show answer

Answer: A — −32-\dfrac{3}{2}

Perpendicular lines have slopes that are negative reciprocals, so the slope of line kk is −32-\dfrac{3}{2}.

Question 2

In the standard (x,y)(x,y) coordinate plane, a line contains the points (3,−2)(3,-2) and (7,10)(7,10). What is the slope of that line?

  1. A13\dfrac{1}{3}
  2. B2
  3. C3
  4. D4
Show answer

Answer: C — 3

The slope is 10−(−2)7−3=124=3\dfrac{10-(-2)}{7-3}=\dfrac{12}{4}=3.

Question 3

A line in the standard (x,y)(x,y) coordinate plane crosses the xx-axis at 4 and the yy-axis at -2. Which of the following is an equation of this line?

  1. Ay=−12x−2y=-\dfrac{1}{2}x-2
  2. By=2x−2y=2x-2
  3. Cy=12x−2y=\dfrac{1}{2}x-2
  4. Dy=12x+2y=\dfrac{1}{2}x+2
Show answer

Answer: C — y=12x−2y=\dfrac{1}{2}x-2

The line passes through (0,−2)(0,-2) and (4,0)(4,0), giving a slope of 0−(−2)4−0=12\dfrac{0-(-2)}{4-0}=\dfrac{1}{2}. Its equation is therefore y=12x−2y=\dfrac{1}{2}x-2.

Practice 62 Coordinate geometry — distance, midpoint, slope, line equations questions in the app

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