Question

(2)

Answer 1

You are correct in saying that the figures are similar; however, the ratio you picked is incorrect. The ratio is 8:4.8 which represents the ratio of JK to PQ

Notice how JK and PQ are the first two letters of JKML and PQRS respectively. So they correspond to one another. Furthermore, notice that 8/4.8 = 1.667 approximately. Also, KM/QR = 2.5/1.5 = 1.667 approximately as well. KM and QR are the two middle letters of JKML and PQRS respectively. The fact we get 1.667 approximately both times, and because of the congruent angles, proves the figures to be similar. 

In short, the actual answer is choice B

Answer 2

For polygons to be similar, they must have the same angles and the ratio of the lengths of the corresponding sides must be the same.

First you should rotate the second polygon 90 degrees clockwise (so angle S is on the left) to make it easier to visualize corresponding sides. Now with the shapes in the same orientation, it seems they could be similar. But we have to verify their angles and lengths.

If the two polygons are similar, then the similarity statement might be JKML~PQRS. Notice the order the angles are listed are the same for both shapes (e.g. top left angle is listed first for both shapes). That being said, you could also have written a similarity statement LMKJ ~ SRQP, or another combination of letters with corresponding angles listed in the same order.

First let's check the corresponding sides.
The ratio between corresponding sides JK and PQ is JK/PQ = 8/4.8 = 1.667
The ratio between corresponding sides JL and PS JL/PS = 2.5/1.5 = 1.667
LM/SR and KM/QR also equal 1.667.
Corresponding sides are proportional. We are halfway there.

Now let's check angles. Angle J equals 115 degrees, and its corresponding angle P also equals 115 degrees, so Angle J = Angle P. 

The other angles are also similar. Quick proof here: draw a diagonal from angle L to K and from S to Q to split the shape to two triangles. The left triangles are similar by SAS similarity and the right triangles are congruent by SSS similarity. Similar triangles have the same angles, so the two entire polygons have the same angles.

Since the polygons JKML and PQRS have the proportional lengths and same angles, they are similar polygons.

Now, the second part in the answer choices is the ratio between JKML and PQRS. In our example of JK and PQ we found the ratio to be JK/PQ = 8/4.8 = 1.667, and every corresopnding side to also be 1.667. So the entire polygon ratio of JKML to PQRS has the ratio 8/4.8.