Lesson 2 Is It Right? Solidify Understanding

Jump Start

The given line segment in the coordinate grid, 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.

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

Open Up the Math: Launch, Explore, Discuss

In Leaping Lizards, you probably thought a lot about parallel and perpendicular lines, particularly when you translated the anchor points of the lizard the same distance and in the same direction, or rotated the lizard about a given center through a $90 °$ angle, or reflected the lizard across a line. It would be helpful to be able to predict when lines in a coordinate grid are parallel or perpendicular to each other.

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?”

You may have written your statements about parallel lines using words such as, “The line through the points $A$ and $B$ is parallel to the line through the points $H$ and $G$ .” We can state this same idea symbolically.

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 Leaping Lizards you 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 and name 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 and name 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 $A (2, 3)$ and $B (0, 6)$ 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 $A (2, 3)$ , $B (0, 6)$ , 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?

6.

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

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 know how to identify perpendicular lines on a coordinate grid, we can also use symbolic notation to indicate that the two lines are perpendicular.

Ready for More?

For experiments 1–4, 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?

Takeaways

When working with lines on a coordinate grid,

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

Adding Notation, Vocabulary, and Conventions

I can indicate two lines are parallel using the following notation:

In representing parallel lines on a coordinate grid, we have drawn images to represent the following undefined terms: point, line, and plane. These are abstract ideas, rather than concrete objects because:

Unlike a dot, a point
Unlike a straight stroke of ink, a line
Unlike a piece of paper, a plane

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 notation for indicating that lines are parallel or perpendicular in our written work.

Retrieval

1.

Use the coordinate grid to find the length of each side of the triangle.

2.

Solve each equation for the indicated variable.

a.

$2 (x + 3) - 1 = 12 x$ (solve for $x$ )

b.

$2 (x + y) + 4 = 12 x$ (solve for $y$ )