Perpendicular lines are two lines that intersect at a 90-degree angle. In other words, if you were to place a protractor at the point of intersection, the angle formed would measure 90 degrees.

- Perpendicular lines have slopes that are negative reciprocals of each other. If the slope of one line is m, then the slope of the perpendicular line is -1/m.
- The product of the slopes of two perpendicular lines is -1.
- Perpendicular lines form right angles at their point of intersection.

Consider the following examples:

Example 1:

Line 1: y = 2x + 3

Line 2: y = -1/2x + 5

To determine if the two lines are perpendicular, we can compare their slopes. Line 1 has a slope of 2, and Line 2 has a slope of -1/2. We can see that these slopes are negative reciprocals of each other, so the lines are perpendicular.

Example 2:

Line 3: 3x - 4y = 8

Line 4: 4x + 3y = 6

To determine if the two lines are perpendicular, we can solve for the slopes of each line. Line 3 can be rewritten as y = (3/4)x - 2, and Line 4 can be rewritten as y = (-4/3)x + 2. Again, we can see that the slopes are negative reciprocals of each other, so the lines are perpendicular.

Key points to remember when studying perpendicular lines:

- Understand the concept of perpendicular lines and how to identify them based on their slopes.
- Be able to find the slope of a line given its equation, and understand how to determine if two lines are perpendicular based on their slopes.
- Practice solving problems involving perpendicular lines to solidify your understanding of the concept.
- Understand the geometric interpretation of perpendicular lines, and how they form right angles at their point of intersection.

Remember to practice identifying perpendicular lines and working with their slopes to become comfortable with the concept.

.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.