The absolute values of the slope and the tangent of the related angle are equal.
Practice makes perfect
a
Let's consider the points given in the exercise and plot on them on a coordinate plane.
We are asked to draw triangles EAB and ECD with E at the origin. Recall that if a point is at the origin, it means that its coordinates are (0,0). Let's add E onto the plane.
We can now connect the points representing the vertices to form â–³ EAB and â–³ ECD.
b
We need to find the values of the tangent of ∠BEA and the tangent of ∠DEC. First, recall that in a right triangle the tangent of an acute angle θ is the ratio between the lengths of an opposite side and a side adjacent to the angle.
tan θ = Opp/Adj
Let's now take a look at the angle ∠BEA. The side opposite the angle is AB and the side adjacent to the angle is AE.
In the picture we see that AB= 6 and AE= 4. Since we know the lengths of the segments, we can now evaluate the tangent of ∠BEA.
We are asked to find the slopes of BE and ED. First, recall that we can calculate the slope of a segment by substituting the coordinates of its endpoints (x_1,y_1) and (x_2,y_2) into the Slope Formula.
m = y_2-y_1/x_2-x_1Let's focus on finding the slope of BE, where B=(-4,6) and E=(0,0). Note that it does not matter which point we choose to use for ( x_1, y_1) and ( x_2, y_2), since the result of using the Slope Formula will be the same.
The slope of BE is - 1.5. Let's now move on to the slope of ED. This time the endpoints are D=(6,8) and E=(0,0). Once again we will substitute them into the Slope Formula and evaluate.
Now it is time for us to make some conclusions about the relationship between slope and tangent. Notice that in both cases the absolute values of the slope and the tangent of the related angle are the same.
tan ∠BEA
1.5
tan ∠DEC
4/3
Slope of BE
- 1.5
Slope of ED
4/3
The reason for that is that the slope of a segment and a tangent of an angle are calculated in a similar way. The opposite side represents vertical change, and the adjacent side represents the horizontal change.
tan θ = Opp/Adj ⇒ tan θ = Δ y/Δ x
We can also see that in the graph.
Now, if we recall the definition of a slope, we know that it is the ratio between the vertical change and the horizontal change.
m = Δ y/Δ x
As a result, the values are the same. The only difference is that a slope of a line or a segment can be negative, unlike the tangent of an acute angle. The tangent of an acute angle cannot be negative, as it is a ratio between lengths of segments.
