# Let z be the angle between the line y=mx+b and x axis. Using the definition of tan z explain why m = tan z

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### 1 Answer

The line `y=mx+b` intersects the x-axis at `((-b)/m,0)` . Let (x,mx+b) be another point on the line, with `x>(-b)/m` .

Drop a perpendicular segment from (x,mx+b) to the x-axis. If the line's x-intercept is labeled A, the point (x,mx+b) labeled B, and the point where the perpendicular segment intersects the x-axis is C then `Delta ABC` is a right triangle. Let the angle at A be `alpha` ; this is the angle between the line and the x-axis.

The geometric definition of the tangent of an acute angle in a right triangle is the ratio of the leg opposite the angle to the leg adjacent to the angle.

Thus `tanalpha=(mx+b-0)/(x-((-b)/m))`

`=(mx+b)/(x+b/m)`

`=(m(mx+b))/(m(x+b/m))`

`=(m(mx+b))/(mx+b)`

`=m`

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