Prove the sides of the polygon similar, and prove the angles congruent

Question

On the same set of axes, graph the image of MNOP for a dilation with center (0, 0) and scale factor 2. Use coordinate geometry and the definition of similar polygons to prove that MNOP is similar to its image. For the graph just make a graph with points O(0,0) P(1,3) M(3,2) N(3,0) Prove the sides of the polygon similar, and prove the angles congruent

Answer 1

for the sides of the polygon to be similar, the corresponding sides must be proportional. This part is pretty straight-forward once you draw the two images and calculate the lengths of the corresponding sides. By definition of dilation by a scale factor of 2, you already know ahead of time that the ratio of the sides will be two. hope this helps!

Answer 2

To prove that MNOP is similar to its image under the dilation, you would need to establish two things:

1) all corresponding sides are proportional

2) all corresponding angles are congruent

To prove proportional sides, you will need to utilize the distance formula for each side of MNOP and its image, and establish that all pairs of sides adhere to a 1: 2 ratio

The prove the congruent angles, the easiest way would be to use slopes of the two sides at each vertex. Please note, this method only works if you have no rotation or reflection of the transformed image. If there is a rotation or reflection, you will need to use more advanced methods involving trigonometry.