prove the angles of the polygons congurent and prove the sides proportional

Question

Copy the graph at the right. 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 congurent

Answer 1

For a dilation through the origin by a factor of k, the image of a point P(x, y) is given by P'(kx,ky).

So find the images of M, N, O, and P by multiplying coordinates by the scale factor 2.

To prove the corresponding sides proportional (here is one example):

1. Determine distance from P to M using distance formula
2. Determine distance from P' to M'
3. Determine the ratio P'M'/PM. Should reduce to the scale factor.
4. Do the same for the other sides of the original quadrilateral and its image. The ratio of corresponding sides all reduce to 2.

To prove corresponding angles congruent one way is to find slopes of corresponding segments showing those slopes are equal. That means corresponding segments are parallel, Now use corresponding angles are congruent when parallel lines cut by a transversal.

See the following graph for details. You can use the slider on k to see different dilations for the original quadrilateral MNOP.

desmos.com/calculator/jb3c9nnjmh