A perpendicular bisector is a line, segment, or ray that intersects another line segment at a 90-degree angle and divides it into two equal parts.

- It always passes through the midpoint of the line segment.
- It forms right angles with the line it bisects.
- It divides the line segment into two equal parts.

To find the perpendicular bisector of a line segment, follow these steps:

- Locate the midpoint of the line segment.
- Draw a line that passes through the midpoint and is perpendicular to the given line segment.

Given the line segment AB, with A(2, 4) and B(6, 8), find the perpendicular bisector of AB.

First, find the midpoint of AB: ( (2+6)/2, (4+8)/2 ) = (4, 6).

Next, the slope of AB = (8-4) / (6-2) = 4/4 = 1. The negative reciprocal of 1 is -1. So, the slope of the perpendicular bisector = -1.

Using the point-slope form, the equation of the perpendicular bisector passing through (4, 6) is: y - 6 = -1(x - 4) or y = -x + 10.

Therefore, the equation of the perpendicular bisector of AB is y = -x + 10.

When studying perpendicular bisectors, make sure to understand the following concepts:

- Definition and properties of perpendicular bisector.
- Finding the midpoint of a line segment.
- Using the midpoint to find the equation of the perpendicular bisector.
- Understanding the relationship between slope and perpendicular lines.

Practice drawing perpendicular bisectors and finding their equations for different line segments to reinforce your understanding of the concept.

Understanding perpendicular bisectors is important as it is a fundamental concept in geometry and plays a significant role in various geometric constructions and proofs.

.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.