A line segment is a part of a line that is bounded by two distinct end points and contains every point on the line between its end points. Here are some key concepts and properties to understand about line segments:

**End Points:** The two points that define the boundaries of a line segment.

**Length:** The distance between the two end points of a line segment. This can be calculated using the distance formula: *d = √((x2 - x1)^2 + (y2 - y1)^2)*

**Collinearity:** All the points on a line segment lie on the same line.

**Midpoint:** The point that divides a line segment into two equal parts. The coordinates of the midpoint can be found using the midpoint formula: *((x1 + x2)/2, (y1 + y2)/2)*

**Bisector:** A line, ray, or segment that divides another segment into two equal parts. The segment joining the midpoint of a line segment is called its bisector.

1. Find the length of the line segment with end points A(3, 4) and B(7, 8).

**Solution:** Using the distance formula, we get: *d = √((7 - 3)^2 + (8 - 4)^2) = √(4^2 + 4^2) = √(16 + 16) = √32*

2. Find the midpoint of the line segment with end points P(2, 3) and Q(8, 5).

**Solution:** Using the midpoint formula, we get: *((2 + 8)/2, (3 + 5)/2) = (5, 4)*

1. Find the length of the line segment with end points C(1, 2) and D(9, 6).

2. Find the midpoint of the line segment with end points E(4, -3) and F(10, 7).

3. Given a line segment with end points G(3, 8) and H(11, 2), find the coordinates of a point that divides the segment into a 2:1 ratio.

.Study GuideSimilarity and scale Worksheet/Answer key

Similarity and scale Worksheet/Answer key

Similarity and scale Worksheet/Answer key

Similarity and scale Worksheet/Answer keyUsing Similar Polygons Worksheet/Answer keySimilar Polygons Worksheet/Answer keyUsing Similar Polygons Worksheet/Answer keySimilar Polygons

Number and Operations (NCTM)

Understand numbers, ways of representing numbers, relationships among numbers, and number systems.

Understand and use ratios and proportions to represent quantitative relationships.

Compute fluently and make reasonable estimates.

Develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equivalent ratios.

Geometry (NCTM)

Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships.

Understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects.

Create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship.

Apply transformations and use symmetry to analyze mathematical situations.

Describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides, and scaling.

Measurement (NCTM)

Apply appropriate techniques, tools, and formulas to determine measurements.

Solve problems involving scale factors, using ratio and proportion.

Grade 8 Curriculum Focal Points (NCTM)

Geometry and Measurement: Analyzing two- and three-dimensional space and figures by using distance and angle

Students use fundamental facts about distance and angles to describe and analyze figures and situations in two- and three-dimensional space and to solve problems, including those with multiple steps. They prove that particular configurations of lines give rise to similar triangles because of the congruent angles created when a transversal cuts parallel lines. Students apply this reasoning about similar triangles to solve a variety of problems, including those that ask them to find heights and distances. They use facts about the angles that are created when a transversal cuts parallel lines to explain why the sum of the measures of the angles in a triangle is 180 degrees, and they apply this fact about triangles to find unknown measures of angles. Students explain why the Pythagorean Theorem is valid by using a variety of methods - for example, by decomposing a square in two different ways. They apply the Pythagorean Theorem to find distances between points in the Cartesian coordinate plane to measure lengths and analyze polygons and polyhedra.