Using similar triangles to explain slope :
In similar triangles, corresponding angles are congruent and corresponding sides are proportional. We can use similar triangles to show that the slope of a line is constant.
Step 1 :
Draw a line ℓ that is not a horizontal line. Label four points on the line as D, F, A, and C.
Step 2 :
Draw the rise and run for the slope between points D and F. Label the intersection as point E. Draw the rise and run for the slope between points A and C. Label the intersection as point B.
We need to show that the slope between the points D and F is the same as the slope between the points A and C.
Step 3 :
Write expressions for the slope between D and F and between A and B.
Slope between D and F : FE / DE
Slope between A and B : CB / AB
Step 4 :
Extend DE and AB across our drawing. DE and AB are both horizontal lines, so they are parallel.
Line l is a transversal that intersects parallel lines.
Step 5 :
Because DE and AB are parallel lines and ℓ is a transversal that intersects DE and AB,
m∠FDE and m∠CAB are corresponding angles and they are congruent.
m∠FED and m∠CBA are right angles and they are congruent.
Step 6 :
By Angle–Angle Similarity, triangle ABE and triangle CDF are similar triangles.
Step 7 :
Because triangle ABE and CDF are similar, the lengths of corresponding sides of similar triangles are proportional.
FE / CB = DE / AB
Step 8 :
Recall that you can also write the proportion so that the ratios compare parts of the same triangle :
FE / DE = CB / AB
Step 9 :
The proportion we wrote in step 8 shows that the ratios we wrote in step 3 are equal. So, the slope of line ℓ is constant.
Suppose that we label two other points on line ℓ as P and Q. Would the slope between these two points be different than the slope we found in the above activity ? Explain.
The slope of the line is constant, so the slope between the points P and Q would be the same. Moreover, not only the two points P and Q, between any two points on ℓ, the slope would be same.
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