Midpoint And Distance Formula Worksheet With Answers

Midpoint and Distance Formula Worksheet: A Comprehensive Guide with Answers

This comprehensive guide provides a detailed explanation of the midpoint and distance formulas, along with numerous practice problems and their solutions. Mastering these formulas is crucial for success in algebra and geometry, forming a foundational understanding for more advanced concepts. We'll break down the formulas, explore their applications, and provide a worksheet with answers to help solidify your understanding.

Understanding the Midpoint Formula

The midpoint formula calculates the coordinates of the point exactly halfway between two given points on a coordinate plane. Imagine you have two points, A and B, with coordinates (x₁, y₁) and (x₂, y₂) respectively. The midpoint, M, lies precisely in the middle of the line segment connecting A and B.

The formula is elegantly simple:

M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

This means you add the x-coordinates of the two points, divide by 2, and similarly add the y-coordinates and divide by 2. The resulting ordered pair represents the coordinates of the midpoint.

Example 1: Finding the Midpoint

Let's say we have point A(2, 4) and point B(6, 10). Using the midpoint formula:

M = ((2 + 6)/2, (4 + 10)/2) = (8/2, 14/2) = (4, 7)

Therefore, the midpoint between A and B is (4, 7).

Example 2: Finding a Coordinate Given the Midpoint

The midpoint formula can also be used to find a missing coordinate. Suppose we know the coordinates of one endpoint, A(1, 3), and the midpoint, M(4, 6). We need to find the coordinates of the other endpoint, B(x₂, y₂).

We can set up two equations using the midpoint formula:

(1 + x₂)/2 = 4 and (3 + y₂)/2 = 6

Solving for x₂ and y₂:

1 + x₂ = 8 => x₂ = 7 3 + y₂ = 12 => y₂ = 9

Therefore, the coordinates of point B are (7, 9).

Understanding the Distance Formula

The distance formula calculates the length of a line segment connecting two points on a coordinate plane. This is essentially an application of the Pythagorean theorem.

The formula is:

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

Where 'd' represents the distance between the two points (x₁, y₁) and (x₂, y₂).

Example 3: Calculating Distance

Let's find the distance between points A(1, 2) and B(4, 6).

d = √[(4 - 1)² + (6 - 2)²] = √[3² + 4²] = √(9 + 16) = √25 = 5

The distance between points A and B is 5 units.

Example 4: Applications of the Distance Formula

The distance formula has numerous applications, including:

• Finding the perimeter of a polygon: Calculate the distance between each pair of consecutive vertices to find the length of each side, then sum them.
• Determining if points are collinear: If the sum of the distances between two pairs of points equals the distance between the remaining pair, the points are collinear.
• Calculating the radius of a circle: The distance between the center of the circle and any point on the circle is the radius.

Midpoint and Distance Formula Worksheet

Now, let's put your knowledge to the test with a practice worksheet. Remember to show your work!

Part 1: Midpoint Formula

1. Find the midpoint of the line segment connecting A(-2, 5) and B(4, 1).
2. Find the midpoint of the line segment connecting C(0, -3) and D(6, 9).
3. One endpoint of a line segment is A(3, -1), and the midpoint is M(5, 2). Find the coordinates of the other endpoint.
4. One endpoint of a line segment is B(-4, 7), and the midpoint is M(1, 0). Find the coordinates of the other endpoint.
5. The midpoint of a line segment is (2, 1). One endpoint is (-1, 4). What are the coordinates of the other endpoint?

Part 2: Distance Formula

1. Find the distance between A(2, 1) and B(5, 5).
2. Find the distance between C(-3, 2) and D(1, -2).
3. Find the distance between E(0, 0) and F(4, 3).
4. Find the distance between G(-2, -5) and H(3, 1).
5. Determine if the points A(1, 2), B(4, 6), and C(7, 10) are collinear.

Part 3: Combined Problems

1. Find the midpoint of the line segment connecting A(-1, 3) and B(5, 7). Then, find the distance between A and the midpoint.
2. A circle has its center at (2, 3) and passes through the point (5, 7). What is the radius of the circle?
3. Find the perimeter of a triangle with vertices at A(1, 1), B(4, 5), and C(7, 2).

Midpoint and Distance Formula Worksheet: Answers

Part 1: Midpoint Formula

1. (1, 3)
2. (3, 3)
3. (7, 5)
4. (6, -7)
5. (5, -2)

Part 2: Distance Formula

1. 5
2. 5.66 (approximately)
3. 5
4. 7.28 (approximately)
5. Yes, they are collinear (check by showing that the sum of distances AB and BC equals AC).

Part 3: Combined Problems

1. Midpoint: (2, 5); Distance between A and the midpoint: 5
2. Radius: 5
3. Perimeter: 13.42 (approximately) (This requires calculating each side length separately using the distance formula and summing them.)

Conclusion

This worksheet and guide offer a comprehensive introduction to the midpoint and distance formulas. Consistent practice is key to mastering these fundamental concepts. Remember to review the formulas, work through the examples, and utilize the solutions provided to identify areas where you might need further review. With diligent practice, you will confidently apply these formulas to solve a wide range of geometry problems. Understanding these formulas is a significant step towards a strong foundation in mathematics. Remember to always double-check your calculations and consider utilizing online calculators to verify your answers, especially when dealing with more complex equations. This consistent practice will ensure mastery of these crucial mathematical tools.