Corresponding lengths and congruent angles
Initial explanation
We have two rectangles, the rectangle of the length is transformed to the rectangle of the right.
Both rectangles are similar, this is, they have the same form but different size. We can see it clearly if we rotate the second rectangle:
Congruent angles
Since all the angles are right angles, all the angles are congruent:
∠A ≅ ∠B ≅ ∠C ≅ ∠D ≅ ∠E ≅ ∠F ≅ ∠G ≅ ∠H
The pairs corresponding angles are between rectangles are:
∠A ≅ ∠G
∠B ≅ ∠H
∠C ≅ ∠F
∠D ≅ ∠E
Constant of proportionality
Now, we can write the corresponding sides
AB ⇄ GH
CD ⇄ FE
AC ⇄ GF
BD ⇄ HE
If we divide each pair of corresponding lengths, we will obtain the same result:
[tex]\frac{AB}{GH}=\frac{CD}{FE}=\frac{AC}{GF}=\frac{BD}{HE}[/tex]
Let's calculate any of those divisions:
[tex]\frac{CD}{FE}=\frac{12}{9}=\frac{4}{3}[/tex]
Then, the constant of proportionality is 4/3
[tex]\frac{AB}{GH}=\frac{CD}{FE}=\frac{AC}{GF}=\frac{BD}{HE}=\frac{4}{3}[/tex]
