Let's draw the two right isosceles triangles and identify the corresponding sides.
Identifying Congruent Angles
Since both triangles are right isosceles triangles, we know that its interior angles are 45-45-90. Therefore, we can tell that there are three pairs of congruent angles.
∠ A ≅ ∠ D
∠ B ≅ ∠ E
∠ C ≅ ∠ F
Comparing the Ratios
Looking at the triangles, we see that we are missing some lengths, so we can use the Pythagorean theorem to find them. In an isosceles triangle, the length of the hypotenuse is given by a formula.
h=sqrt(2)* l
Here, l is the leg of the triangle. Let's substitute 1 for the leg in the above formula to find the length of the missing side of the △ DEF.
h=sqrt(2)* (1) ⇔ h=sqrt(2)
Now, we must find the value of the leg for the △ ABC. Let's substitute 3 for the hypotenuse of △ ABC in the above formula and solve for l.
Therefore, the missing length of △ ABC is 3sqrt(2). With this, we have the values of all the sides of both triangles, we are going to model the given triangles again with the found values and corresponding side lengths.
Now, let's compare the ratios of the corresponding sides. Recall that the included side between a pair of angles of one polygon corresponds to the included side between the corresponding pair of congruent angles of another polygon.
Corresponding Sides
Ratio
Substitute Lengths
Simplify
A B, D E
AB/DE
3/sqrt(2)/1
3/sqrt(2)
B C, E F
BC/EF
3/sqrt(2)
3/sqrt(2)
C A, F D
CA/FD
3/sqrt(2)/1
3/sqrt(2)
As we can see, all of the ratios are equal. Therefore, given polygons are similar. However, we can see that we have a radical in the denominator. Hence, we must rationalize the denominator. To do this, we will multiply the numerator and denominator by sqrt(2).
