Finding a diagonal in a trapezoid given the other diagonal and three sides The figure below is a trapezoid, what is the length of the red line?
Thank you very much in advance!


Well, I think I see a method of proceeding.  Let $x$ be one of the 2 equal angles of the isoceles triangle to the left.  It's possible to use the law of cosines to calculate $\cos x$.
It turns out the left triangle is congruent to the right triangle.  At least if you assume the drawing is accurate enough that the top and bottom are the 2 parallel lines of the trapezoid and not left and right.  The 2 equal angles of the left and right isoceles triangles are equal due to alternate interior angles (had to look up the name, been a while).  So the third angle of the triangle on the right is $180^o-2x$ and the lower right angle of the trapezoid is $x+180^o-2x=180^o-x.$
Now you can use the law of cosines along with $\cos(180^o-x)=-\cos x$ to solve for the length of the red line.
Not sure it's as elegant as dot dot's answer, but let's make sure it gets the same result.  We get that
$$\cos x=\frac{49^2+28^2-49^2}{2\times49\times28}=\frac27$$
$$c^2=49^2+28^2-2\times49\times28\times-\frac27=$$
$$49\times7^2+16\times7^2+16\times7^2=81\times7^2$$
$$c=9\times7=63$$
