Since LJ∥ PM we do not have to do any rotation for KL and LJ to lie upon PN and PM, respectively.
See solution
Practice makes perfect
An isosceles right triangle has one angle that is 90^(∘) and two angles that are congruent. If we call the measures of these angle m∠ a, we can set up the following equation:
m∠ a+m∠ a=90^(∘)
Let's solve this equation.
Thus, both triangles have two 45^(∘) angles and one right angle. This means they have the same shape and are therefore similar.
Prove by mapping
We can also prove that the figures are similar by mapping, for example, the smaller triangle onto the bigger. Let's translate △ JKL so that the vertices at the triangles right angles maps onto each other. Since a translation is a rigid motion, it preserves length and angle measures which means KL will lie on NP and LJ will lie on PM. Additionally, no rotations are needed as LJ ∥ MP.
From the diagram we see that PN=v and L'K'=t. By dividing the lengths of these corresponding sides, we can determine the scale factor we need when dilating.
Scale Factor: v/t
So, using L' as the center of dilation, we can dilate △ J'K'L to the same size as △ MNP if we multiply its sides with a scale factor of vt.
L'K':& v/t* t=v [0.8em]
L'J':& v/t* t=v
Having dilated △ J''K''L'' with a scale factor of vt, we can see that △ K''J''L'' maps onto △ MNP proving they are similar.
