Given rectangle ABCD with K the midpoint AD and A D / A B = 2 , find the angle between BK and diagonal AC.

Question

Given rectangle ABCD with K the midpoint AD and   A  D      /    A  B  =      2   , find the angle between BK and diagonal AC.

Kyler Crawford · Accepted Answer

Step 1Since   A  K  =            A      D        2   we have:            A      B              A      K        =  2            A      B              A      D        =      2  And you can proceed to show that the triangles ABK and ACD are similar (they are both right-angled). From which tracing equal angles gives you your answer.Trigonometric ratios are (Amongst other things) a convenient way of encoding similarity. Sometimes pure geometry is simpler.

tuehanhyd8ml · Accepted Answer

Step 1The angle between BK and AC is the difference in the angles each makes with the x axisAngle of BK is       ϕ    1    =      tan          −      1        ⁡      (                  1                              2                                /                    2                      )    =      tan          −      1        ⁡      (          2        )  Angle of AC is       ϕ    2    =  π  −      tan          −      1        ⁡      (                  1                  2                      )  Hence the angle between them is  ϕ  =      ϕ    2    −      ϕ    1    =  π  −      tan          −      1        ⁡      (                  1                  2                      )    −      tan          −      1        ⁡      (          2        )  Taking the tangent of the angle   ϕ , we get  tan  ⁡  (  ϕ  )  =  −                                          1                          2                                      +                  2                            1        −                                            2                                      2                                            =  undefinedSince the tangent of the angle ϕ is undefined, then it follows that   ϕ  =      90    ∘   .

Given rectangle ABCD with K the midpoint AD and

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