### Video Transcript

If π΄π΅πΆπ· is similar to πΈπΉπΊπ», find the scale factor of similarity of πΈπΉπΊπ» to π΄π΅πΆπ· and the values of π and π.

In our image, we have two quadrilaterals, one π΄π΅πΆπ· and another πΈπΉπΊπ». These two quadrilaterals are similar. In similar polygons, corresponding angles are congruent and corresponding sides are proportional. Notice here angle π· and angle π» are congruent as are πΆ and πΊ. And because both of these shapes are quadrilaterals, we know that their interior angles sum to 360 degrees. Each of them have three congruent angles, which means their fourth angle will also be congruent.

This confirms that vertices π΄, π΅, πΆ, and π· correspond to the vertices πΈ, πΉ, πΊ, and π», respectively, as given in the question. We know then that side length π΄π΅ corresponds to side length πΈπΉ. Side length π΅πΆ corresponds to side length πΉπΊ. Side length πΆπ· corresponds to side length πΊπ». And side length π·π΄ corresponds to side length π»πΈ.

If two shapes are similar, it means that their corresponding sides are all in the same proportion, and we can express this using similarity ratios. For example, the ratio of corresponding sides between the shapes are equal. The ratio of side length π΄π΅ over πΈπΉ will be equal to π΅πΆ over πΉπΊ, which is also equal to πΆπ· over πΊπ» and π·π΄ over π»πΈ. We can also express the corresponding relationships of side lengths within each quadrilateral. So, the ratio of side length π΄π΅ over π·πΆ would be equal to πΈπΉ over π»πΊ. For similar polygons, we have these similarity ratios.

We could take these similarity ratios and find the values of π and π. However, weβre being asked to find something called the scale factor of similarity. We use scale factors to multiply lengths in one of the shapes to work out corresponding lengths in the other shape. But we need to be a bit careful about how we define and describe our scale factors to make sure we are clear about our start and end points for the calculations. It works like this. If we take a side length from the polygon π΄π΅πΆπ· and multiply it by the scale factor, weβll find the corresponding side lengths in πΈπΉπΊπ». In this case, weβre going from a larger shape to a smaller shape, in which case we would expect the scale factor to be less than one.

On the other hand, thereβs a different scale factor if weβre going to take a side length from the polygon πΈπΉπΊπ» and to find the corresponding side length in π΄π΅πΆπ·. In this case, weβre taking a side length from the smaller polygon and trying to find the corresponding side length in the larger polygon. And we expect that the scale factor will be greater than one. Our job here in deciding the scale factor of similarity is deciding are we trying to work from the larger to the smaller or from the smaller to the larger. At first glance of πΈπΉπΊπ» to π΄π΅πΆπ· seems to imply weβre taking the smaller polygon πΈπΉπΊπ» and trying to find the larger π΄π΅πΆπ·.

Upon closer inspection, the similarity of πΈπΉπΊπ» to π΄π΅πΆπ· implies that the shape π΄π΅πΆπ· is the original shape and πΈπΉπΊπ» is the shape that is similar to it. The scale factor for this similarity is the multiplier that we should apply to the lengths in π΄π΅πΆπ· to find their corresponding lengths in πΈπΉπΊπ». That was the first option we considered, where we expected to have a scale factor of less than one. Since we have the values of two corresponding side lengths between π΄π΅πΆπ· and πΈπΉπΊπ», we can calculate the scale factor. If we take π΅πΆ from triangle π΄π΅πΆπ· and the corresponding side πΊπΉ in πΈπΉπΊπ», and we let π be our unknown scale factor, π΅πΆ equals 47. πΉπΊ equals 18.8.

If we divide both sides of the equation by 47, we get a scale factor of 0.4. Itβs more common to write scale factors in fraction form, so we have a scale factor of two-fifths. Letβs use this information to find π and π. π΄π΅ corresponds to πΈπΉ, and therefore π plus four times two-fifths equals 19.2. Dividing both sides by two-fifths, we find π plus four equals 48. Subtracting four from both sides and we find π equals 44.

To find the value of π, we have the corresponding side π·πΆ multiplied by the scale factor of two-fifths equals π»πΊ. 34 times two-fifths equals π, which makes π 13.6. The final answer then, scale factor equals two-fifths, π equals 13.6, and π equals 44.