Midpoint Formula: 5 Essential Worksheet Answers

If you're here, you're likely looking to enhance your understanding or help someone else with the midpoint formula in geometry. Whether you're a student trying to pass your next exam or a curious learner exploring the intricacies of coordinate geometry, understanding the midpoint formula is fundamental. Here, we'll break down the 5 essential worksheet answers that will strengthen your grasp of this concept.

What is the Midpoint Formula?

Before we delve into the worksheet answers, let’s first understand what the midpoint formula is. The midpoint formula calculates the exact middle point between two given points (x₁, y₁) and (x₂, y₂) in a coordinate system. The formula is:

[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) ]

Worksheet Problem 1: Finding Midpoint of Two Points

• Question: Find the midpoint of the line segment joining the points (4, 2) and (6, 8).
• Answer: Using the midpoint formula, we get:

X-Coordinate	Y-Coordinate
(4 + 6)/2 = 5	(2 + 8)/2 = 5

Thus, the midpoint is (5, 5).

🔔 Note: The midpoint will always lie on the straight line segment connecting the two points.

Worksheet Problem 2: Verifying Midpoints

• Question: Given point A(2, 1) and midpoint M(4, -1), find the coordinates of point B.
• Answer: Let (x, y) be the coordinates of point B. We know:

[ \left( \frac{2 + x}{2} , \frac{1 + y}{2} \right) = (4, -1) ]

Solving for x and y:

\[ 2 + x = 8 \Rightarrow x = 6 \] \[ 1 + y = -2 \Rightarrow y = -3 \]

Therefore, the coordinates of point B are (6, -3).

Worksheet Problem 3: Real-World Application

• Question: If two surveyors are at positions A(-2, 3) and B(6, 5), where would the midpoint be if they were to mark the middle point of their distance?
• Answer: Using the midpoint formula:

[ \left( \frac{-2 + 6}{2} , \frac{3 + 5}{2} \right) ]

This calculates to (2, 4).

⚠️ Note: Midpoints are not just theoretical; they have practical applications in land surveying, GPS navigation, and computer graphics.

Worksheet Problem 4: Understanding Slopes and Midpoints

• Question: If two points have a slope of -1, and one point is (3, 7), what is the midpoint between this point and another point?
• Answer: Let’s call the other point (x, y). We know the slope is:

[ \frac{y - 7}{x - 3} = -1 ]

After solving for x and y:

\[ x = 3 - y + 7 \Rightarrow x = -y + 10 \]

We can't solve for the exact midpoint without another point, but knowing the slope tells us the line's direction. The midpoint's x-coordinate will be on a line parallel to the given point with a slope of -1.

Worksheet Problem 5: Midpoint and Distance

• Question: Given points A(1, 1) and B(7, 9), find the midpoint and the distance between them.
• Answer:

  Calculating the midpoint:

  [ \left( \frac{1 + 7}{2} , \frac{1 + 9}{2} \right) ]

  This gives us (4, 5) as the midpoint.

  Now, finding the distance:

  [ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ] [ = \sqrt{(7 - 1)^2 + (9 - 1)^2} ] [ = \sqrt{36 + 64} ] [ = \sqrt{100} = 10 ]

  The distance between points A and B is 10 units.

The midpoint formula is more than a mathematical tool; it’s a concept that spans across numerous applications in real life and mathematics. Whether you're navigating a map, designing a video game, or simply calculating the center of a garden, the midpoint formula helps you find the balance point. We've explored various scenarios through our worksheet answers, from basic calculations to applications in real-world contexts, ensuring a comprehensive understanding of the topic. Remember, mastering this formula involves not only computing midpoints but also understanding its implications in different geometric settings.