Similar Triangles as Slope
Common Core Standard: Use similar triangles to explain why the slope m is the same between any two distinct points on a
non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the
equation y = mx + b for a line intercepting the vertical axis at b.
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Similar triangles - are triangles that have angles with the same measure and have proportional side lengths
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Example:
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Between any two points on a line, you can draw a triangle
The line will form the hypotenuse (longest side) of the triangle
All you need to do is draw up from the bottom most point until at the same height as the next point, then draw
horizontally until you touch the point, thus making a triangle
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Example:
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Using these triangles, we can determine the slope of the line
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Remember slope is the change in y over the change in x
So, count how many units the triangle is tall for change in y and count how many units the triangle is wide
for change in x
Example:
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On this line, there are three triangles, but we are only going to use the small one (green)
Notice, the green triangle includes the points on the line (0,0) and (3, 2)
To get from (0,0) to (3,2), there is a vertical line that is 2 units tall and there is a horizontal line that is 3
units to create the triangle
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Thus, the slope is
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The y-intercept is (0, 0)
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The equation is y =
x