1 2 Practice Line Segments And Distance Answer Key

1-2 Practice: Line Segments and Distance – A Comprehensive Guide with Answer Key

This comprehensive guide delves into the fundamentals of line segments and distance, providing a thorough understanding of the concepts, accompanied by solved examples and an answer key for the practice problems. Understanding line segments and distance is crucial for building a solid foundation in geometry and related mathematical fields. This article will equip you with the necessary tools and knowledge to confidently tackle problems involving line segments and distance calculations.

Understanding Line Segments

A line segment is a part of a line that is bounded by two distinct endpoints. Unlike a line, which extends infinitely in both directions, a line segment has a definite beginning and end. We often represent line segments using capital letters corresponding to their endpoints. For example, the line segment with endpoints A and B is denoted as AB or BA (the order doesn't matter in this case).

Key Properties of Line Segments:

• Length: The most important characteristic of a line segment is its length, which represents the distance between its two endpoints. The length is always a positive value.
• Midpoint: The midpoint of a line segment is the point that divides the segment into two equal parts.
• Collinearity: If multiple points lie on the same line, they are said to be collinear. Line segments are inherently collinear.

Measuring the Distance Between Two Points

The distance between two points is the length of the line segment connecting them. The method for calculating this distance depends on the coordinate system being used.

Distance in One Dimension (Number Line):

In a one-dimensional coordinate system (a number line), the distance between two points A and B with coordinates x_A and x_B respectively, is simply the absolute difference between their coordinates:

Distance = |x_B - x_A| = |x_A - x_B|

The absolute value ensures the distance is always positive, regardless of the order of the points.

Example:

Find the distance between points A and B on a number line, where A is at -3 and B is at 5.

Solution:

Distance = |5 - (-3)| = |5 + 3| = 8

Distance in Two Dimensions (Coordinate Plane):

In a two-dimensional coordinate system (the Cartesian plane), we use the distance formula derived from the Pythagorean theorem. For two points A(x₁, y₁) and B(x₂, y₂), the distance d between them is:

Distance (d) = √[(x₂ - x₁)² + (y₂ - y₁)²]

Example:

Find the distance between points A(2, 3) and B(6, 7).

Solution:

d = √[(6 - 2)² + (7 - 3)²] = √[(4)² + (4)²] = √(16 + 16) = √32 = 4√2

Practice Problems: Line Segments and Distance

Now let's put our knowledge into practice with some example problems. Remember to show your work clearly and use the appropriate formulas.

Problem 1:

Points A, B, and C are collinear. A is at -2, B is at 4, and C is at 10. Find the lengths of AB, BC, and AC.

Problem 2:

Find the distance between points P(1, -2) and Q(5, 4) on a coordinate plane.

Problem 3:

The midpoint M of line segment AB has coordinates (3, 5). If point A has coordinates (1, 2), find the coordinates of point B.

Problem 4:

Three points have coordinates: A(-1, 2), B(3, 0), and C(5, 6). Determine if these points are collinear. (Hint: Check if the distance between A and C equals the sum of the distances between A and B, and B and C).

Problem 5:

A rectangle has vertices at A(1, 2), B(5, 2), C(5, 6), and D(1, 6). Find the length of the diagonal AC.

Problem 6:

Two points are located at (-3, 4) and (7, -2). Find the midpoint of the line segment connecting these points.

Problem 7:

A line segment has endpoints (x, 8) and (4, -2). If the midpoint is (1, 3), find the value of x.

Problem 8:

Find the distance between points (-5, 12) and (3, -8).

Answer Key:

Problem 1:

• AB = |4 - (-2)| = 6
• BC = |10 - 4| = 6
• AC = |10 - (-2)| = 12

Problem 2:

d = √[(5 - 1)² + (4 - (-2))²] = √(16 + 36) = √52 = 2√13

Problem 3:

Let B = (x, y). The midpoint formula gives: ( (1+x)/2, (2+y)/2 ) = (3, 5)

Solving for x and y: (1+x)/2 = 3 => x = 5; (2+y)/2 = 5 => y = 8. Therefore, B = (5, 8).

Problem 4:

AB = √[(3 - (-1))² + (0 - 2)²] = √(16 + 4) = √20 BC = √[(5 - 3)² + (6 - 0)²] = √(4 + 36) = √40 AC = √[(5 - (-1))² + (6 - 2)²] = √(36 + 16) = √52

Since AB + BC = √20 + √40 ≠ √52 = AC, the points are not collinear.

Problem 5:

AC = √[(5 - 1)² + (6 - 2)²] = √(16 + 16) = √32 = 4√2

Problem 6:

Midpoint = ( (-3 + 7)/2, (4 + (-2))/2 ) = (2, 1)

Problem 7:

Midpoint = ( (x+4)/2, (8 + (-2))/2 ) = (1, 3)

(x+4)/2 = 1 => x = -2

Problem 8:

d = √[(3 - (-5))² + (-8 - 12)²] = √(64 + 400) = √464 = 4√29

This comprehensive guide provides a detailed explanation of line segments and distance, accompanied by solved practice problems. Mastering these concepts is essential for further studies in geometry and related fields. Remember to practice regularly to build your understanding and problem-solving skills. Further exploration into advanced geometric concepts can build upon this foundational knowledge.