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.

House transformation

Thank you in advance!

  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.


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