Lesson 3 Make It Right Solidify Understanding

Jump Start

The given line segment in the coordinate grid below with endpoints at $A (2, 6)$ and $B (6, 2)$ is one side of a triangle. Describe the triangle formed if each of the following points is the third vertex $C$ of the triangle. (Use words in your description such as: acute, obtuse, right, scalene, isosceles, equilateral.)

1.

If $C (2, 4)$ is the third vertex of the triangle then the triangle is

2.

If $C (2, 2)$ is the third vertex of the triangle then the triangle is

3.

If $C (0, 0)$ is the third vertex of the triangle then the triangle is

4.

If $C (6, 6)$ is the third vertex of the triangle then the triangle is

5.

If $C (3, 2)$ is the third vertex of the triangle then the triangle is

6.

If $C (1, 2)$ is the third vertex of the triangle then the triangle is

Learning Focus

Recognize parallel and perpendicular lines in a coordinate plane.

Write the equation of a line parallel or perpendicular to a given line through a given point.

How can I determine if two lines in a coordinate plane are parallel or perpendicular? Is visual examination (that is, “It looks like it is”) sufficient?

Open Up the Math: Launch, Explore, Discuss

1.

Examine the following diagram and write statements to describe which lines are parallel to each other.

My list of parallel lines:

How did you know that these lines were parallel, other than, “They look like parallel lines?”

Now that we have made an observation about the slopes of parallel lines, it will be helpful to make an observation about the slopes of perpendicular lines. Perhaps in previous work you have used a protractor or some other tool or strategy to help you make a right angle. In this task we consider how to create a right angle by attending to slopes on the coordinate grid.

We begin by stating a fundamental idea for our work: Horizontal and vertical lines are perpendicular. For example, on a coordinate grid, the horizontal line $y = 2$ and the vertical line $x = 3$ intersect to form four right angles.

But what if a line or line segment is not horizontal or vertical? How do we determine the slope of a line or line segment that will be perpendicular to it?

2.

Experiment 1

a.

Consider the points $A (2, 3)$ and $B (4, 7)$ and the line segment, $\overset{―}{A B}$ , between them. What is the slope of this line segment?

b.

Locate a third point $C (x, y)$ on the coordinate grid, so the points $A (2, 3)$ , $B (4, 7)$ , and $C (x, y)$ form the vertices of a right triangle, with $\overset{―}{A B}$ as its hypotenuse.

c.

Explain how you know that the triangle you formed contains a right angle.

d.

Now rotate this right triangle $90 °$ about the vertex point $A (2, 3)$ . Explain how you know that you have rotated the triangle $90 °$ .

e.

Compare the slope of the hypotenuse of this rotated right triangle with the slope of the hypotenuse of the pre-image. What do you notice?

3.

Experiment 2

a.

Consider the points $A (2, 3)$ and $B (5, 4)$ , and the line segment, $\overset{―}{A B}$ , between them. What is the slope of this line segment?

b.

Locate a third point $C (x, y)$ on the coordinate grid, so the points $A (2, 3)$ , $B (5, 4)$ and $C (x, y)$ form the vertices of a right triangle, with $\overset{―}{A B}$ as its hypotenuse.

c.

Explain how you know that the triangle you formed contains a right angle.

d.

Now rotate this right triangle $90 °$ about the vertex point $A (2, 3)$ . Explain how you know that you have rotated the triangle $90 °$ .

e.

Compare the slope of the hypotenuse of this rotated right triangle with the slope of the hypotenuse of the pre-image. What do you notice?

4.

Experiment 3

a.

Consider the points $A (2, 3)$ and $B (7, 5)$ , and the line segment, $\overset{―}{A B}$ , between them. What is the slope of this line segment?

b.

Locate a third point $C (x, y)$ on the coordinate grid, so the points $A (2, 3)$ , $B (7, 5)$ , and $C (x, y)$ form the vertices of a right triangle, with $\overset{―}{A B}$ as its hypotenuse.

c.

Explain how you know that the triangle you formed contains a right angle.

d.

Now rotate this right triangle $90 °$ about the vertex point $A (2, 3)$ . Explain how you know that you have rotated the triangle $90 °$ .

e.

Compare the slope of the hypotenuse of this rotated right triangle with the slope of the hypotenuse of the pre-image. What do you notice?

5.

Experiment 4

a.

Consider the points $A (0, 0)$ and $B (a, b)$ , and the line segment, $\overset{―}{A B}$ , between them. What is the slope of this line segment?

b.

Locate a third point $C (x, y)$ on the coordinate grid, so the points $A (0, 0)$ , $B (a, b)$ and $C (x, y)$ form the vertices of a right triangle, with $\overset{―}{A B}$ as its hypotenuse.

c.

Explain how you know that the triangle you formed contains a right angle.

d.

Now rotate this right triangle $90 °$ about the vertex point $A (0, 0)$ . Explain how you know that you have rotated the triangle $90 °$ .

e.

Compare the slope of the hypotenuse of this rotated right triangle with the slope of the hypotenuse of the pre-image. What do you notice?

6.

Based on experiments 1–4, state an observation about the slopes of perpendicular lines.

Observation:

While this observation is based on a few specific examples, can you create an argument or justification for why this is always true?

Now that we have made observations about the slopes of parallel and perpendicular lines, we can use these observations to write equations of lines that are parallel or perpendicular to a given line or line segment through a given point.

7.

Given the line segment whose endpoints are the points $A (4, 3)$ and $B (8, 9)$ , write the equations of the following lines.

(Try to do this without sketching the graphs of segment $A B$ or the lines described. Then check your work by sketching your results on the coordinate grid.)

a.

Write the equation of the line perpendicular to segment $A B$ through its midpoint.

b.

Write the equation of the line parallel to segment $A B$ through the point $(1, 5)$ .

c.

Write the equation of the line perpendicular to segment $A B$ through the point $(1, 5)$ .

d.

Check your work by sketching your results on the coordinate grid.

Ready for More?

For experiments 1–4 described earlier in the lesson, write the equations of the lines that contain the hypotenuse of the original triangle and the hypotenuse of the rotated triangle in both point-slope form and slope-intercept form, using the point at the center of rotation for the point-slope form of the equation. What relationships do you notice in these equations?

Experiment 1:	Original hypotenuse	Rotated hypotenuse
Point-slope form
Slope-intercept form

Experiment 2:	Original hypotenuse	Rotated hypotenuse
Point-slope form
Slope-intercept form

Experiment 3:	Original hypotenuse	Rotated hypotenuse
Point-slope form
Slope-intercept form

Experiment 4:	Original hypotenuse	Rotated hypotenuse
Point-slope form
Slope-intercept form

Takeaways

When working with lines on a coordinate grid,

I know the lines are parallel if
I know the lines are perpendicular if

Vocabulary

Lesson Summary

In this lesson, we learned criteria for determining if two lines in a coordinate plane are parallel or perpendicular. We also learned how to write equations of lines parallel or perpendicular to a given line through a given point.

Retrieval

1.

Fill in the graphic organizer with the quadrilateral that has the listed characteristics.

2.

Use the two points to write the equation of the line in standard form that goes through the points.

$(- 6, 20)$ and $(3, - 7)$