Example Solution: A 180^(∘) rotation about the origin followed by a reflection in the line x=2.
Practice makes perfect
To describe a congruence transformation that maps â–³ DEF to â–³ JKL, let's begin by plotting the triangles with the given vertices.
The two triangles have different orientations and locations. In △ JKL, the vertex J is above K and L and in △ DEF the vertex D is below E and F. Therefore, if we rotate △ DEF 180^(∘) about the origin, the figures will have the same orientation. We can do it using the coordinate rule.
preimage (a,b) → image (- a,- b)When a point is rotated 180^(∘) about the origin, its image has opposite coordinates. Let's find the coordinates of the image.
(a,b)
(- a,- b)
D(- 3,- 4)
D'(3,4)
E(-5,-1)
E'(5,1)
F(-1,1)
F'(1,-1)
The image â–³ D'E'F' does not have the same coordinates as â–³ JKL. Let's plot the triangle, to find what transformation should we perform now.
It seems that there is some kind of symmetry. Let's check for the vertical line x=2.
Indeed, x=2 forms an axis of symmetry, so we can reflect â–³ D'E'F' across the line.
We can see that the figures mapped onto each other. Therefore, the congruence transformation that maps △ DEF to △ JKL is a 180^(∘) rotation about the origin followed by a reflection in the line x=2.
