Triangle ABC has been rotated 90° to create triangle DEF. Using the image below, prove that lines BC and EF have the opposite and reciprocal slopes. You must show all of your work to receive credit.

Answer 1

First, assume that line BC has a slope

Slope of line BC = tan(θ) ---------------(1)

Then, line EF has rotated 90^o or π/2 and its slope is given by

Slope of line EF = tan(θ + π/2 ) -------(2)

Now, we could re-write the tangent function in terms of sine and cosine as follows:

tan(θ + π/2 ) = sin (θ + π/2 ) / cos( θ + π/2 ) ----(3)

We then use the properties of sin and cos of sum of angles

sin (θ + π/2) = sin(θ) cos(π/2) + cos(θ) sin(π/2) ----(4)

cos (θ + π/2) :

= cos(θ) cos(π/2) - sin(θ) sin(π/2) ----(5)

Recall that

cos(π/2) = 0

sin(π/2) = 1

Then, substituting in equations (4) and (5) by cos(π/2) = 0 and sin(π/2) = 1, we get

sin (θ + π/2) = sin(θ) (0) + cos(θ) (1) = cos(θ) ----(6)

cos (θ + π/2) = cos(θ) (0) - sin(θ) 1) = - sin(θ) ----(7)

Thus, equation (2) becomes

Slope of line EF = tan(θ + π/2 ) = sin (θ + π/2) / sin (θ + π/2)

= cos(θ) / - sin(θ)

= - cot (θ) ----------------------(8)

Thus, multiplying the slopes of BC (equation (1)) and EF,(equation (8)) , we get

Slope of line BC x Slope of line EF = tan(θ) x tan(θ + π/2 )

= - tan(θ) x cot (θ )

= -1