1 3 Additional Practice Midpoint And Distance Answer Key

13 Additional Practice: Midpoint and Distance Formula - Answers & Explanations

This comprehensive guide provides answers and detailed explanations for 13 additional practice problems focusing on the midpoint and distance formulas. Mastering these formulas is crucial for success in geometry and related fields. We'll cover various scenarios, including finding midpoints, determining distances, and applying these concepts to solve real-world problems. Remember, understanding the why behind the formulas is as important as knowing the formulas themselves.

Understanding the Midpoint Formula

The midpoint formula allows us to find the coordinates of the point exactly halfway between two given points. If we have two points, (x₁, y₁) and (x₂, y₂), the midpoint (xₘ, yₘ) is calculated using the following formula:

xₘ = (x₁ + x₂) / 2

yₘ = (y₁ + y₂) / 2

This essentially averages the x-coordinates and the y-coordinates separately to find the midpoint.

Example Problem 1:

Find the midpoint of the line segment connecting A(2, 5) and B(8, 1).

Solution:

Using the midpoint formula:

xₘ = (2 + 8) / 2 = 5

yₘ = (5 + 1) / 2 = 3

Therefore, the midpoint is (5, 3).

Understanding the Distance Formula

The distance formula helps us calculate the length of a line segment given the coordinates of its endpoints. For two points (x₁, y₁) and (x₂, y₂), the distance (d) between them is:

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

This formula is derived from the Pythagorean theorem. It essentially finds the lengths of the horizontal and vertical legs of a right-angled triangle formed by the two points and then uses the Pythagorean theorem to find the hypotenuse, which represents the distance between the points.

Example Problem 2:

Find the distance between points C(-3, 2) and D(4, -1).

Solution:

Using the distance formula:

d = √[(4 - (-3))² + (-1 - 2)²] = √[(7)² + (-3)²] = √(49 + 9) = √58

Therefore, the distance between C and D is √58.

13 Additional Practice Problems with Detailed Solutions

Let's now dive into the 13 additional practice problems. Each problem will be presented, followed by a step-by-step solution.

Problem 3: Find the midpoint of the line segment with endpoints E(-6, 4) and F(2, -8).

Solution:

xₘ = (-6 + 2) / 2 = -2

yₘ = (4 + (-8)) / 2 = -2

Midpoint: (-2, -2)

Problem 4: Find the distance between points G(0, 5) and H(12, 9).

Solution:

d = √[(12 - 0)² + (9 - 5)²] = √(144 + 16) = √160 = 4√10

Problem 5: The midpoint of a line segment is (3, -1). One endpoint is (1, 5). Find the other endpoint.

Solution: Let the other endpoint be (x, y). Using the midpoint formula:

3 = (1 + x) / 2 => x = 5

-1 = (5 + y) / 2 => y = -7

Other endpoint: (5, -7)

Problem 6: Find the distance between points I(-2, -7) and J(6, 1).

Solution:

d = √[(6 - (-2))² + (1 - (-7))²] = √(64 + 64) = √128 = 8√2

Problem 7: Find the midpoint of the line segment with endpoints K(10, -2) and L(-4, 6).

Solution:

xₘ = (10 + (-4)) / 2 = 3

yₘ = (-2 + 6) / 2 = 2

Midpoint: (3, 2)

Problem 8: Determine the distance between the points M(√2, 3) and N(√8, 7). Simplify your answer.

Solution:

d = √[(√8 - √2)² + (7 - 3)²] = √[(2√2 - √2)² + 4²] = √[(√2)² + 16] = √(2 + 16) = √18 = 3√2

Problem 9: The midpoint of a line segment is (-4, 0). One endpoint is (2, -6). Find the other endpoint.

Solution: Let the other endpoint be (x, y).

-4 = (2 + x) / 2 => x = -10

0 = (-6 + y) / 2 => y = 6

Other endpoint: (-10, 6)

Problem 10: Find the distance between points O(a, b) and P(c, d).

Solution:

d = √[(c - a)² + (d - b)²] This is the general distance formula.

Problem 11: A rectangle has vertices at R(1, 1), S(5, 1), T(5, 4), and U(1, 4). Find the length of the diagonal RT.

Solution: Use the distance formula on points R and T:

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

Problem 12: Find the midpoint of the line segment connecting points V(-3.5, 2.1) and W(1.5, -0.9).

Solution:

xₘ = (-3.5 + 1.5) / 2 = -1

yₘ = (2.1 + (-0.9)) / 2 = 0.6

Midpoint: (-1, 0.6)

Problem 13: Two points are 10 units apart. One point is at (2, 3). If the x-coordinate of the second point is 8, find the possible y-coordinates of the second point.

Solution: Let the second point be (8, y). Using the distance formula:

10 = √[(8 - 2)² + (y - 3)²]

100 = 36 + (y - 3)²

(y - 3)² = 64

y - 3 = ±8

y = 11 or y = -5

Possible y-coordinates: 11 and -5

Advanced Applications and Further Practice

The midpoint and distance formulas are foundational concepts in geometry. They are used extensively in more advanced topics like:

• Coordinate Geometry Proofs: Proving geometric properties (e.g., parallelogram, rhombus) using coordinates and the distance and midpoint formulas.
• Circles: Finding the equation of a circle given its center and radius (utilizing the distance formula).
• Vectors: The midpoint formula can be extended to vectors in higher dimensions.

To further solidify your understanding, try creating your own problems using different coordinate points and varying levels of complexity. Practice is key to mastering these important concepts. Remember to always double-check your calculations and carefully interpret the results within the geometric context of the problem. By consistently applying these formulas and understanding their underlying principles, you’ll build a strong foundation in coordinate geometry and related mathematical fields.