Can You Show That △ DEF~△ JKL by Using the AA Similarity Theorem? Yes. Congruent Corresponding Angles: ∠ D≅∠ J, ∠ E≅∠ K, ∠ F≅∠ L Similarity Transformation: (x,y)→(- 2x,- 2y)
Practice makes perfect
We will start by drawing △ DEF and △ JKL on the same coordinate plane. To do so we will plot the given points and then connect them to form two triangles.
If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.
We have one pair of congruent angles. To use the AA Similarity Theorem, we have to find another pair of congruent angles. Let's take a look at our diagram.
We can see that ∠ E and ∠ K are right angles. In case you want to see the proof of this statement, it is included below. We have found a second pair of congruent angles, ∠ E≅∠ K. Therefore by AA Similarity Theorem, △ DEF~△ JKL. Let's list the corresponding congruent angles.
∠ D≅∠ J
∠ E≅∠ K
∠ F≅∠ L
Finally, we will write the similarity transformation that maps △ DEF to △ JKL. Let's consider all possible similarity transformations.
A dilation is a transformation in which a figure is enlarged or reduced with respect to a fixed point.
We have to determine which of these transformations were used to map △ DEF to △ JKL. Let's take a look at our diagram.
We can see that DF is located in the second quadrant. It corresponds to JL, which is located in the fourth quadrant. This suggests a rotation counterclockwise about the origin of 180^(∘). Recall that the coordinate rule for such rotation can be written as (x,y)→ (- x,- y).
Triangle JKL appears to be twice as large as triangle D'E'F'. Therefore, the second transformation is a dilation centered at the origin by a scale factor of 2. Recall that the coordinate rule for such dilation can be written as (x,y)→(2x,2y).
We will write the coordinate rule for the composition of similarity transformations that maps △ DEF to △ JKL. Recall that the first transformation is (x,y)→( -x, -y) and the second transformation is (x,y)→( 2x, 2y). Let's write the composition of these two transformations.
(x,y)→( -x, -y)→( - 2x, - 2y)
⇕
(x,y)→(- 2x,- 2y)
Note that a composition of similarity transformations is also a similarity transformation.
Showing Our Work
Proof
In case you want to see the proof that ∠ E and ∠ K are right angles, we have included our reasoning. Two line segments are perpendicularif and only if the product of their slopes is - 1. To calculate the slope m of a line segment through points ( x_1, y_1) and ( x_2, y_2), we used the Slope Formula.
m=y_2- y_1/x_2- x_1
We calculated the slopes of the segments DE and EF. Then we checked if their product is equal to - 1, which is equivalent to ∠ E being a right angle. Recall that D(- 8,5), E(- 5,8), and F(- 1,4).
Since m_(DE)* m_(EF)=- 1, the line segments DE and EF are perpendicular. In other words, ∠ E is a right angle. We followed a similar process to show that the line segments JK and KL are perpendicular, which is equivalent to ∠ K being a right angle.
