What you should learn

To determine if two lines are parallel or perpendicular by their slopes

To write equations of lines that pass through a given point, parallel or perpendicular to the graph of a given equation.

NCTM Curriculm Standards 2, 3, 6 - 10

 

In doing this the teacher wants to make sure that the following words are incorporated into the introductory lesson:

Perpendicular Lines

Parallel Lines

Parallelogram

Introduction: Kite-making has been a national sport in some Far East countries since ancient times. In September, Chinese families celebrate Kites' Day. Entire families go outside and fly kites. These kites may be simple diamond shapes or more complex shapes like fish, birds, dragons, and colorfully-dressed people.

The outline of a simple kite is shown below on the coordinate plane. The ordered pairs for the tipes of the kite are labeled. We know that the two slats that support the kite meet at right angles. Lines that intersect at right angles are called perpendicular lines.

Lines in the smae plane that never intersect are called parallel lines. Parallel lines can form families of graphs. In Lesson 6-5A, you learned that families of graphs can either have the same slope or the same intercept. The graph below shows a family of graphs represented by the following equations.

y = 2x

y = 2x + 3

y = 2x - 1

The slope of each line is 2. Notice that the lines seem to be parallel. In fact, they are parallel.

Defintion of Parallel Lines in a Coordinate Plane: If two nonvertical lines have the same slope, then they are parallel. All vertical lines are parallel.

 

A parallelogram is a quadrilateral in which opposite sides are parallel. We can use slope to determine if quadrilaterals graphed on a coordinate plane are parallelograms.

Exercise 1: Determine whether quadrilateral ABCD is a parallelogram if its vertices are A (-5, -3), B (5,3), C (7, 9), and D (-3, 3). (Use your problem solving skills and plot the points and find the slope of the lines, if opposite lines have equal slopes, the figure is a parallelogram).

You can write the equation of a line parallel to another line if you know a point on the line and the equation of the other.

Exercise 2: Write an equation in slope-intercept form of the line that passes through (4, 0) and is parallel to the graph of 4x - 3y = 2.

Sometimes graphs can be drawn in a way that leads you to the wrong conclusion. Mathematics can help you determine if a graph is misleading.

We have seen that the slopes of parallel lines are equal. What is the relationship of slopes of perpendicular lines?

Exercise 3: Refer to the application at the beginning of the lesson.

a. Write equations of the lines containing the slats of the kite in slope-intercept form.

b. Determine the relationship between the slopes of perpendicular lines.

 

 

These resluts will suggest the following definition.

Definition of Perpendicular Lines in a Coordinate Plane: If the product of the slopes of two lines is -1, then the lines are perpendicular. In a plane, vertical lines and horizontal lines are perpendicular.

 

You can use a right triangle and a coordinate grid to model the slopes of perpendicular lines.

You can use your knowledge of perpendicular lines to write equations of lines perpendicular to a given line.

Exercise 4: Write the slope-intercept form of an equation that passes through (8, -2) and is perpendicular to the graph of 5x - 3y = 7.

First find the slope of the given line.

5x - 3y = 7

-3y = -5x + 7

y = 5x/3 - 7/3

The slope of the line is 5/3. The slope of the line perpendicular to this line is the negative reciprocal of 5/3, or -3/5.

Use the point-slope form to find the equation.

y - y1 = m (x - x1)

y - (-2) = (-3/5)(x - 8)

y + 2 = -3x/5 + 24/5

y = -3x/5 + 14/5

An equation for the line is y = (-3/5)x + 14/5

Activity: Perpendicular Lines on a Coordinate Plane

Materials: Grid paper and scissors

a. A scalene triangle is one in which no two sides are equal. Cut out a scalene right triangle ABC so that C is the right angle. Label the vertices and the sides as shown below.

b. Draw a coordinate plane on the grid paper. Place ABC on the coordinate plane so that A is at the origin and side b lies along the positive x-axis.

c. Name the coordines of B.

d. What is the slope of side c?

YOUR TURN

a. Rotate the triangle 90 degrees counterclockwise so that A is still at the origin and side b is along the positive y-axis.

b. Name the coordinates of B.

c. What is the slope of c?

d. What is the relationship between the first position of c and the second?

e. What is the relationship between the slopes of in each position?

Closing Activity: Check for understanding by using this as a quick review before class is over. It should take about the last five to ten minutes. I would use it for my students as their 'ticket out the door'. Click Here.

Homework: The homework to be assigned for tonight would be: 19 - 45 odd, 46, 47, 49 - 58

Alternative Homework: Enriched: 18 - 44 even, 46 - 58

Extra Practice: Students book page 771 Lesson 6-6

Extra Practice Worksheet: Click Here.

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