Midpoint questions show up at least once per test on the ACT® Math. A midpoint is the point exactly in the middle of a line segment, and it's a concept that appears frequently on the test. I'll show you what midpoints are, how to calculate them, and how to tackle midpoint questions on the ACT®.
What is a Midpoint?
The midpoint of a line segment is the point that divides it into two equal parts. It's like finding the middle of a ruler - it's equidistant from both endpoints. On the ACT®, you'll often need to calculate midpoints or use them to solve more complex problems.
Calculating Midpoints
To find the midpoint, I use the average of the x-coordinates and the average of the y-coordinates of the two endpoints. Here's a simple formula:
Midpoint = ( ,
Where (x₁, y₁) is the first point and (x₂, y₂) is the second point.
Practice Questions
Let's tackle some practice questions to help you master midpoints for the ACT®.
Question 1:
Find the midpoint of the line segment with endpoints A(3, 7) and B(11, -5).
- A) (7, 1)
- B) (8, 2)
- C) (5, 1)
- D) (6, 4)
- E) (7, 4)
Click for the Answer
Correct Answer: A) (7, 1)
Explanation:
Using the midpoint formula: ((3 + 11)/2, (7 + (-5))/2) = (14/2, 2/2) = (7, 1)
Question 2:
Find the midpoint of the line segment with endpoints C(-2, 8) and D(4, -2).
- A) (1, 3)
- B) (2, 3)
- C) (1, 2)
- D) (1, -2)
- E) (3, 1)
Click for the Answer
Correct Answer: A) (1, 3)
Explanation:
Using the midpoint formula: ((-2 + 4)/2, (8 + (-2))/2) = (2/2,6/2) = (1, 3)
How Midpoints Appear on the ACT®
On the ACT®, you'll encounter midpoint questions in various forms. Sometimes you'll be given coordinates and asked to calculate the midpoint, like in the examples above. Other times, you might need to find the midpoint by reading points off a graph. Let's look at an example of a graph-based question:
Question 3:
Find the midpoint of the line segment with endpoints A and B shown in the graph above.
- A) (6, 5)
- B) (7, 5)
- C) (6, 6)
- D) (5, 5)
- E) (6, 4)
Click for the Answer
Correct Answer: A) (6, 5)
Explanation:
From the graph, we can see that point A is at (2, 8) and point B is at (10, 2). Using the midpoint formula: ((2 + 10)/2, (8 + 2)/2) = (12/2, 10/2) = (6, 5)
Key Takeaways
Finding midpoints on the ACT® is all about averaging the x-coordinates and y-coordinates of the endpoints. Whether you're given coordinates or a graph, the process remains the same.