ACT Math Practice Test: Coordinate Geometry
- Coordinate Geometry
- 10 questions
- 12 minutes
These 10 questions cover the coordinate geometry that appears throughout the ACT Math section: slopes, equations of lines, distance, circles, parabolas and transformations. A quick sketch of the points often shows which choice is reasonable before you calculate.
Tap an option to mark your answer. You can change your mind, or tap the same option again to leave the question blank. The correct answers and explanations appear when you finish the test.Tap an option to answer; tap it again to leave the question blank.
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Question 1
What is the slope of the line 3x − 4y = 12 in the standard (x, y) coordinate plane?
Correct answer: B
Solve for y: −4y = −3x + 12, so y = (3/4)x − 3 and the slope is 3/4. Forgetting that dividing by −4 changes both signs gives −3/4.
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Question 2
Which equation describes the line that passes through the point (2, −1) and is perpendicular to the line y = 2x + 5?
Correct answer: D
A perpendicular line has the negative reciprocal slope, −1/2. Substituting (2, −1): −1 = −1/2 · 2 + b, so b = 0 and y = −(1/2)x. The line y = 2x − 5 passes through the point but is parallel, not perpendicular.
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Question 3
What is the distance, in coordinate units, between the points (−2, 3) and (4, −5)?
Correct answer: C
The horizontal change is 6 and the vertical change is 8, so the distance is the square root of 6² + 8² = 100, which is 10. Adding the changes instead gives 14.
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Question 4
What are the center and the radius of the circle with equation (x − 3)2 + (y + 2)2 = 49?
Correct answer: A
In (x − h)² + (y − k)² = r², the center is (h, k) and the radius is r. Here h = 3, k = −2 and r² = 49, so r = 7. Reading 49 as the radius forgets the square.
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Question 5
What are the coordinates of the vertex of the parabola y = x2 − 6x + 5?
Correct answer: B
The vertex has x = −b/(2a) = 6/2 = 3, and y = 9 − 18 + 5 = −4. Using x = −3 is a sign error and gives (−3, 32).
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Question 6
At what point do the lines y = 2x − 1 and x + y = 8 intersect?
Correct answer: D
Substitute: x + (2x − 1) = 8, so 3x = 9, x = 3 and y = 5. The point (5, 3) is on the second line but not the first, because 2(5) − 1 = 9.
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Question 7
The point (4, −3) is reflected across the y-axis, and the image is then moved 5 units up. What are the coordinates of the final point?
Correct answer: C
Reflecting across the y-axis changes the sign of x: (−4, −3). Moving 5 units up adds 5 to y: (−4, 2). The choice (4, 8) comes from reflecting across the x-axis instead.
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Question 8
The line y = mx + 4 passes through the point (6, 1). What is the value of m?
Correct answer: A
Substitute the point: 1 = 6m + 4, so 6m = −3 and m = −1/2. Getting 1/2 comes from dropping the negative sign when 4 is subtracted from both sides.
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Question 9
Which equation describes the line that is parallel to 2x + 5y = 10 and passes through the point (0, −3)?
Correct answer: D
A parallel line keeps the same left side, 2x + 5y. Substituting (0, −3) gives 2(0) + 5(−3) = −15. The choice 2x + 5y = −3 uses the y-coordinate as the constant; 2x − 5y = 15 contains the point but has slope 2/5, not −2/5.
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Question 10
A triangle has vertices (1, 2), (5, 2) and (5, 8) in the standard (x, y) coordinate plane. What is the perimeter of the triangle, in coordinate units?
Correct answer: B
The horizontal side is 4 and the vertical side is 6. The third side is the square root of 4² + 6² = 52, which is 2√13. The perimeter is 4 + 6 + 2√13 = 10 + 2√13 (about 17.2). The choice 20 adds 4 + 6 + 10, guessing the long side.
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