Lesson 16Writing Equations for Lines

Learning Goal

Let’s explore the relationship between points on a line and the slope of the line.

Learning Targets

  • I can decide whether a point is on a line by finding quotients of horizontal and vertical distances.

Lesson Terms

  • similar
  • slope

Warm Up: Coordinates and Lengths in the Coordinate Plane

Problem 1

Triangle ABE contained within triangle ACD. Points given are: A (0,2), B (2,2), C (4,2), D (4,7)

Find each of the following and explain your reasoning:

  1. The length of segment .

  2. The coordinates of .

Activity 1: What We Mean by an Equation of a Line

Problem 1

Line is shown in the coordinate plane.

Line j is graphed in the coordinate plane with the origin labeled O. The numbers 1 through 10 are indicated on each axis. The line begins in quadrant 3 and slants upward and to the right passing through the point labeled A at zero comma zero, the point labeled B at 4 comma 3, and the point labeled D at 8 comma 6. Point C is also indicated at 4 comma zero.

  1. What are the coordinates of and ?

  2. Is point on line ? Explain how you know.

  3. Is point on line ? Explain how you know.

  4. Is point on line ? Explain how you know.

  5. Suppose you know the - and -coordinates of a point. Write a rule that would allow you to test whether the point is on line .

Activity 2: Writing Relationships from Slope Triangles

Problem 1

Here are two diagrams:

  1. Complete each diagram so that all vertical and horizontal segments have expressions for their lengths.

    Triangle DEF and ABC. Information given: D (0,1), E (x,y). AC=4, BC=3
    Triangle DEF and ABC. Information given: D (2,4), E (x,y). AC=4, BC=3

  2. Use what you know about similar triangles to find an equation for the quotient of the vertical and horizontal side lengths of in each diagram.

Are you ready for more?

Problem 1

A rectangle 10 wide and 6 high. A right triangle is attached on either side, 4 wide and 6 high. An isosceles triangle is attached on the top, 10 wide and 8 high. All 4 shapes form a larger triangle.

  1. Find the area of the shaded region by summing the areas of the shaded triangles.

  2. Find the area of the shaded region by subtracting the area of the unshaded region from the large triangle.

  3. What is going on here?

Lesson Summary

Here are the points , , and on the same line. Triangles and are slope triangles for the line so we know they are similar triangles. Let’s use their similarity to better understand the relationship between and , which make up the coordinates of point .

A line graphed in the x y plane with the origin labeled O. The numbers negative 1 through 6 are indicated on the x axis and the numbers negative 1 through 8 are indicated on the y axis. The line begins in quadrant 3, slants upwards and to right passing through the point zero comma zero which is labeled A, the point one comma 2 which is labeled C, and the point x comma y which is labeled E. Point B is indicated directly below point C at one comma zero and point D is indicated directly below point E at x comma zero.

The slope for triangle is since the vertical side has length 2 and the horizontal side has length 1. The slope we find for triangle is because the vertical side has length and the horizontal side has length . These two slopes must be equal since they are from slope triangles for the same line, and so:

Since this means that the value of is twice the value of , or that . This equation is true for any point on the line!

Here are two different slope triangles. We can use the same reasoning to describe the relationship between and for this point .

A line graphed in the x y plane with the origin labeled O. The numbers negative 1 through 6 are indicated on the x axis and the numbers negative 1 through 8 are indicated on the y axis. The line begins in quadrant 3, slants upwards and to right passing through the point zero comma one which is labeled A, the point one comma 3 which is labeled C, and the point x comma y which is labeled E. Point B is indicated directly below point C at one comma one and point D is indicated directly below point E at x comma one. The distance between point a and point d is indicated by x and the distance between point D and point E is indicated by y minus 1.

The slope for triangle is since the vertical side has length 2 and the horizontal side has length 1. For triangle , the horizontal side has length . The vertical side has length because the distance from to the -axis is but the vertical side of the triangle stops 1 unit short of the -axis. So the slope we find for triangle is . The slopes for the two slope triangles are equal, meaning:

Since is twice , another way to write this equation is . This equation is true for any point on the line!