Question

The triangle whose vertices are (2,5), (3,1) and (4,2) is transformed by the rule (x,y)(x-2,y+4). Is the image similar or congruent to the original figure?

Answer 1

The image is congruent to the original figure.

Step-by-step explanation:

To determine whether the image is similar or congruent to the original figure, we need to analyze the effects of the transformation rule on the original triangle. The rule given is (x,y)(x-2,y+4), which means that each point of the triangle is shifted 2 units to the left and 4 units up. By applying this rule to each vertex of the original triangle, we obtain the new coordinates:

Vertex 1: (2-2, 5+4) = (0, 9)

Vertex 2: (3-2, 1+4) = (1, 5)

Vertex 3: (4-2, 2+4) = (2, 6)

Now, we can compare the lengths of the sides of the original triangle and the image triangle:

Original triangle side lengths:
Side 1: sqrt((3-2)^2 + (1-5)^2) = sqrt(1+16) = sqrt(17)
Side 2: sqrt((4-3)^2 + (2-1)^2) = sqrt(1+1) = sqrt(2)
Side 3: sqrt((2-4)^2 + (5-2)^2) = sqrt(4+9) = sqrt(13)

Image triangle side lengths:
Side 1: sqrt((1-0)^2 + (5-9)^2) = sqrt(1+16) = sqrt(17)
Side 2: sqrt((2-1)^2 + (6-5)^2) = sqrt(1+1) = sqrt(2)
Side 3: sqrt((0-2)^2 + (9-6)^2) = sqrt(4+9) = sqrt(13)

As we can see, the lengths of the sides of the original triangle and the image triangle are equal, which means they are congruent. Therefore, we can conclude that the image is congruent to the original figure.