Geometry Distance and Midpoint Formula Worksheet Answers
Understanding the distance and midpoint formulas is crucial in geometry, as they help you calculate the length between two points and find the middle point of a line segment. In this article, we will delve into the world of geometry, exploring the distance and midpoint formulas, and provide answers to a worksheet on these topics.
Distance Formula
The distance formula is used to calculate the distance between two points in a coordinate plane. The formula is:
Distance = √((x2 - x1)² + (y2 - y1)²)
where (x1, y1) and (x2, y2) are the coordinates of the two points.
📝 Note: The distance formula is derived from the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Midpoint Formula
The midpoint formula is used to find the coordinates of the midpoint of a line segment. The formula is:
Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)
where (x1, y1) and (x2, y2) are the coordinates of the endpoints of the line segment.
📝 Note: The midpoint formula is used to find the point that divides the line segment into two equal parts.
Worksheet Answers
Here are the answers to a worksheet on the distance and midpoint formulas:
Section 1: Distance Formula
- Find the distance between the points (2, 3) and (4, 6).
Answer: √((4 - 2)² + (6 - 3)²) = √(2² + 3²) = √(4 + 9) = √13
- Find the distance between the points (-1, 2) and (3, 4).
Answer: √((3 - (-1))² + (4 - 2)²) = √(4² + 2²) = √(16 + 4) = √20
- Find the distance between the points (0, 0) and (2, 2).
Answer: √((2 - 0)² + (2 - 0)²) = √(2² + 2²) = √(4 + 4) = √8
Section 2: Midpoint Formula
- Find the midpoint of the line segment with endpoints (2, 3) and (4, 6).
Answer: ((2 + 4) / 2, (3 + 6) / 2) = (3, 4.5)
- Find the midpoint of the line segment with endpoints (-1, 2) and (3, 4).
Answer: ((-1 + 3) / 2, (2 + 4) / 2) = (1, 3)
- Find the midpoint of the line segment with endpoints (0, 0) and (2, 2).
Answer: ((0 + 2) / 2, (0 + 2) / 2) = (1, 1)
Section 3: Mixed Questions
- Find the distance between the points (1, 2) and (3, 4), and then find the midpoint of the line segment with these endpoints.
Answer: Distance = √((3 - 1)² + (4 - 2)²) = √(2² + 2²) = √8 Midpoint = ((1 + 3) / 2, (2 + 4) / 2) = (2, 3)
- Find the midpoint of the line segment with endpoints (-2, 3) and (4, 6), and then find the distance between these points.
Answer: Midpoint = ((-2 + 4) / 2, (3 + 6) / 2) = (1, 4.5) Distance = √((4 - (-2))² + (6 - 3)²) = √(6² + 3²) = √(36 + 9) = √45
As you can see, mastering the distance and midpoint formulas is essential in geometry. With practice and patience, you’ll become proficient in using these formulas to solve various problems.
What is the distance formula used for?
+The distance formula is used to calculate the distance between two points in a coordinate plane.
What is the midpoint formula used for?
+The midpoint formula is used to find the coordinates of the midpoint of a line segment.
How are the distance and midpoint formulas related?
+The distance and midpoint formulas are both used to calculate the length and midpoint of a line segment, but they are used in different contexts and have different applications.
Related Terms:
- Midpoint Formula worksheet answer
- Midpoint of a line Worksheet
