Lesson 11

Writing Equations for Lines

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

11.1: Coordinates and Lengths in the Coordinate Plane

X and Y axes.. A line through point A, at 0 comma 2, point E, and point D at 4 comma 7. A line connects E to B at 2 comma 2. Another connects D to C at 4 comma 2.

Find each of the following and explain your reasoning:

  1. The length of segment \(BE\).
  2. The coordinates of \(E\).

11.2: What We Mean by an Equation of a Line

Line \(j\) is shown in the coordinate plane.

  1. What are the coordinates of \(B\) and \(D\)?

  2. Is point \((20,15)\) on line \(j\)? Explain how you know.

  3. Is point \((100,75)\) on line \(j\)? Explain how you know.

  4. Is point \((90,68)\) on line \(j\)? Explain how you know.

  5. Suppose you know the \(x\)- and \(y\)-coordinates of a point. Write a rule that would allow you to test whether the point is on line \(j\).

Coordinate plane, quadrant 1. Line j goes through point A, at the origin, point B at 4 comma 3, point D at 8 comma 6. Dotted lines connect points A, and B to point C at 4 comma 0.

 

11.3: Writing Relationships from Slope Triangles

Here are two diagrams:

  1. Complete each diagram so that all vertical and horizontal segments have expressions for their lengths.
  2. Use what you know about similar triangles to find an equation for the quotient of the vertical and horizontal side lengths of \(\triangle DFE\) in each diagram.
Two triangles. A, at 4 comma 4, B at 8 comma 7, C at 8 comma 4. Horizontal side length 4, vertical side length 3. Next, point D at 0 comma 1, E at x comma y, F at x comma 1.
Two triangles. A, at 6 comma 7, B at 10 comma 10, C at 10 comma 7. Horizontal side length 4, vertical side length 3. Next, point D at 2 comma 4, E at x comma y, F at x comma 4.


  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?
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.

Summary

Here are the points \(A\), \(C\), and \(E\) on the same line. Triangles \(ABC\) and \(ADE\) are slope triangles for the line so we know they are similar triangles. Let’s use their similarity to better understand the relationship between \(x\) and \(y\), which make up the coordinates of point \(E\).

A line graphed in a coordinate plane.

The slope for triangle \(ABC\) is \(\frac{2}{1}\) since the vertical side has length 2 and the horizontal side has length 1. The slope we find for triangle \(ADE\) is \(\frac{y}{x}\) because the vertical side has length \(y\) and the horizontal side has length \(x\). These two slopes must be equal since they are from slope triangles for the same line, and so: \(\frac{2}{1} = \frac{y}{x}\).

Since \(\frac{2}{1} = 2\) this means that the value of \(y\) is twice the value of \(x\), or that \(y= 2x\). This equation is true for any point \((x,y)\) on the line!

Here are two different slope triangles. We can use the same reasoning to describe the relationship between \(x\) and \(y\) for this point \(E\).

A line graphed in a coordinate plane.

The slope for triangle \(ABC\) is \(\frac{2}{1}\) since the vertical side has length 2 and the horizontal side has length 1. For triangle \(ADE\), the horizontal side has length \(x\). The vertical side has length \(y-1\) because the distance from \((x,y)\) to the \(x\)-axis is \(y\) but the vertical side of the triangle stops 1 unit short of the \(x\)-axis. So the slope we find for triangle \(ADE\) is \(\frac{y-1}{x}\). The slopes for the two slope triangles are equal, meaning: \(\displaystyle \frac{2}{1} = \frac{y-1}{x}\)

Since \(y-1\) is twice \(x\), another way to write this equation is \(y-1 = 2x\). This equation is true for any point \((x,y)\) on the line!

Glossary Entries

  • similar

    Two figures are similar if one can fit exactly over the other after rigid transformations and dilations.

    In this figure, triangle \(ABC\) is similar to triangle \(DEF\).

    If \(ABC\) is rotated around point \(B\) and then dilated with center point \(O\), then it will fit exactly over \(DEF\). This means that they are similar.

  • slope

    The slope of a line is a number we can calculate using any two points on the line. To find the slope, divide the vertical distance between the points by the horizontal distance.

    The slope of this line is 2 divided by 3 or \(\frac23\).

    A graph in the coordinate plane with a line on it.