A slope triangle for a line is a triangle whose longest side lies on the line and whose other two sides are vertical and horizontal. This lesson establishes the remarkable fact that the quotient of the vertical side length and the horizontal side length does not depend on the triangle: this number is called the slope of the line. The argument builds on many key ideas developed in this unit:
- The dilation of a slope triangle, with center of dilation on the line, is a slope triangle for the same line.
- Triangles sharing two common angle measures are similar.
- Quotients of corresponding sides in similar polygons are equal.
In future lessons, they will use slope to write equations for lines.
- Comprehend the term “slope” to mean the quotient of the vertical distance and the horizontal distance between any two points on a line.
- Draw a line on a coordinate grid given its slope and describe (orally) observations about lines with the same slope.
- Justify (orally) that all “slope triangles” on one line are similar by using transformations or Angle-Angle Similarity.
Let’s learn about the slope of a line.
If using the print version of the materials, students need a straightedge in order to draw lines. If using the digital version, an applet is made available for this purpose.
- I can draw a line on a grid with a given slope.
- I can find the slope of a line on a grid.
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.
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 \(\frac23\).
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