In geometry, 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. The perpendicular bisector is a fundamental concept in geometry and is commonly used to find the midpoint of a line segment or to construct right angles.

Properties of a Perpendicular Bisector

Here are the key properties of a perpendicular bisector:

It intersects the line segment at a 90-degree angle (right angle).
It divides the line segment into two equal parts.
It passes through the midpoint of the line segment.

Construction of a Perpendicular Bisector

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

Use a compass to measure and mark equal distances from both ends of the line segment.
Draw two arcs of the same radius above and below the line segment.
Connect the intersection points of the arcs with a straight line. This line is the perpendicular bisector of the original line segment.

Example Problem

Find the equation of the perpendicular bisector of the line segment with endpoints A(2, 4) and B(6, 10).

First, find the midpoint of the line segment using the midpoint formula:

Midpoint = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)

Midpoint = ((2 + 6) / 2, (4 + 10) / 2) = (4, 7)

Now, find the slope of the line passing through points A and B:

Slope = (y₂ - y₁) / (x₂ - x₁) = (10 - 4) / (6 - 2) = 6 / 4 = 3/2

The negative reciprocal of the slope is -2/3, which is the slope of the perpendicular bisector. Using the midpoint (4, 7) and the slope -2/3, we can find the equation of the perpendicular bisector line.

Equation of the perpendicular bisector: y - 7 = -2/3(x - 4)

Study Guide

When studying perpendicular bisectors, make sure to focus on the following key points:

Understanding the definition and properties of a perpendicular bisector.
Mastering the construction of a perpendicular bisector using a compass and straightedge.
Practicing finding the midpoint of a line segment and determining the equation of the perpendicular bisector.

Additionally, solving problems involving perpendicular bisectors in different contexts will help reinforce your understanding of the concept.

◂Math Worksheets and Study Guides Seventh Grade. Exploring Area and Surface Area

Study Guide

Exploring Area and Surface Area

Worksheet/Answer key

Exploring Area and Surface Area

Worksheet/Answer key

Exploring Area and Surface Area

Worksheet/Answer key

Exploring Area and Surface Area

Worksheet/Answer key

Exploring Area and Surface Area

The resources above cover the following skills:

Geometry (NCTM)

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

Precisely describe, classify, and understand relationships among types of two- and three-dimensional objects using their defining properties.

Use visualization, spatial reasoning, and geometric modeling to solve problems.

Use geometric models to represent and explain numerical and algebraic relationships.

Measurement (NCTM)

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

Select and apply techniques and tools to accurately find length, area, volume, and angle measures to appropriate levels of precision.

Develop and use formulas to determine the circumference of circles and the area of triangles, parallelograms, trapezoids, and circles and develop strategies to find the area of more-complex shapes.

Develop strategies to determine the surface area and volume of selected prisms, pyramids, and cylinders.

Grade 7 Curriculum Focal Points (NCTM)

Measurement and Geometry and Algebra: Developing an understanding of and using formulas to determine surface areas and volumes of three-dimensional shapes

By decomposing two- and three-dimensional shapes into smaller, component shapes, students find surface areas and develop and justify formulas for the surface areas and volumes of prisms and cylinders. As students decompose prisms and cylinders by slicing them, they develop and understand formulas for their volumes (Volume = Area of base x Height). They apply these formulas in problem solving to determine volumes of prisms and cylinders. Students see that the formula for the area of a circle is plausible by decomposing a circle into a number of wedges and rearranging them into a shape that approximates a parallelogram. They select appropriate two- and three-dimensional shapes to model real-world situations and solve a variety of problems (including multi-step problems) involving surface areas, areas and circumferences of circles, and volumes of prisms and cylinders.