Find rotation angle of given image

Question

At first: our aim is to find the total transformation of left house to the right house.

What I did it first is translating the house with the center to the origin. I already found out that the transformation is done by scaling, then rotating, then shearing and after that doing a translation again.

The issue I have right now is that if I had the rotation angle I could solve all the other transformation matrices - but this is the point where I am stuck. Do you have any idea how the rotation can be resolved? I would be happy for any hint that would lead me to find the angle of the rotation.

Thank you in advance!

Answer 1

1. Label the midpoint at the bottom of the final figure as $A$ and the midpoint of the top of the parallelogram as $B$.
2. Since $\overline{PM} \perp \overline{MA}$, $\triangle{AMP}$ is a right triangle.
3. We're given that the length of the hypotenuse $|AP|=5$ and $|AM| = \displaystyle \frac{|v'_y|}{2}$.
4. Since $\triangle{AMP} \cong \triangle{BMP}$, hypotenuses $|BP| = |AP| = 5$.
5. Because $\angle{APB} = 90^\circ$ the larger $\triangle{APB}$ is also a right triangle
6. Since we know that $|BP| = |AP| = 5, \angle{PAB}=\arctan\left({\frac{5}{5}}\right) =\frac{\pi}{4} = 45^\circ$.

I've skipped some steps, so it's not a rigorous proof, but just let me know if any step is unclear.