Lesson 4-3: Similar Triangles and Slope

Property

Two figures are similar if, and only if:

1. Their corresponding angles are equal, and
2. Their corresponding sides are proportional.

Examples

• A rectangle with sides 4 cm and 6 cm is similar to a rectangle with sides 8 cm and 12 cm. The scale factor is 2.

• A photograph measuring 4 inches by 6 inches is enlarged to 8 inches by 12 inches. The enlarged photo is similar to the original.

Property

If two triangles are similar, then the ratios of their corresponding side lengths are equal.

For similar triangles $\Delta ABC$ and $\Delta DEF$, the proportion of corresponding sides is written as:

Property

To find the slope of a line from its graph:

1. Locate two points on the graph whose coordinates are integers.
2. Starting with the point on the left, sketch a right triangle, going from the first point to the second point.
3. Count the rise and the run on the legs of the triangle.
4. Take the ratio of rise to run to find the slope, $m = \frac{\operatorname{rise}}{\operatorname{run}}$.

Examples

• A line passes through $(1, 2)$ and $(4, 8)$. The rise is $8 - 2 = 6$ and the run is $4 - 1 = 3$. The slope is $m = \frac{6}{3} = 2$.
• A line passes through $(0, 6)$ and $(2, 2)$. The rise is $2 - 6 = -4$ and the run is $2 - 0 = 2$. The slope is $m = \frac{-4}{2} = -2$.
• A line on a graph connects points $(-3, 1)$ and $(5, 7)$. The rise is $7-1=6$ and the run is $5-(-3)=8$. The slope is $m = \frac{6}{8} = \frac{3}{4}$.

Explanation

To find slope from a graph, pick two easy-to-read points. Count the vertical distance (rise) and horizontal distance (run) to get from one point to the other. The slope is simply the rise divided by the run.