Question Video: Verifying Whether the Two Given Polygons Are Similar Mathematics • 8th Grade

Is polygon ๐ด๐ต๐ถ๐ท similar to polygon ๐บ๐น๐ธ๐‘‹?

04:24

Video Transcript

Is polygon ๐ด๐ต๐ถ๐ท similar to polygon ๐บ๐น๐ธ๐‘‹?

From the figure, we can see that the two polygons weโ€™ve been given are each parallelograms. So we can conclude that they are at least the same type of shape to begin with. To determine whether theyโ€™re similar, we need to test two things. Firstly, we need to test whether corresponding pairs of angles are congruent. And secondly, we need to test whether corresponding pairs of sides are in proportion or in the same ratio.

Now, itโ€™s important to remember that when weโ€™re working with similar polygons, the order of the letters is important. So if these polygons are similar, then the angle at ๐ด will correspond to the angle at ๐บ. The angle at ๐ต will correspond to the angle at ๐น, and so on. We can therefore deduce that the polygons have been drawn in the same orientation.

Letโ€™s consider the angles first of all then. In polygon ๐บ๐น๐ธ๐‘‹, weโ€™ve been given a marked angle of 110 degrees. And in polygon ๐ด๐ต๐ถ๐ท, weโ€™ve been given a marked angle of 70 degrees. One thing we do know about parallelograms is that their opposite angles are equal. So in parallelogram ๐บ๐น๐ธ๐‘‹, the angle at ๐‘‹ will be 110 degrees. And in parallelogram ๐ด๐ต๐ถ๐ท, the angle at ๐ถ will be 70 degrees.

Letโ€™s consider the angle at ๐บ in the polygon ๐บ๐น๐ธ๐‘‹. By extending the line ๐น๐บ, we now see that we have two parallel lines ๐‘‹๐บ and ๐ธ๐น and a transversal ๐บ๐น. We know that corresponding angles in parallel lines are equal, which means that the angle above the line ๐‘‹๐บ will be equal to the angle above the line ๐ธ๐น. Itโ€™s 110 degrees. We also know that angles on a straight line sum to 180 degrees, which means the angle below ๐‘‹๐บ will be 180 minus 110. Itโ€™s 70 degrees. This shows us that the angle at ๐บ in polygon ๐บ๐น๐ธ๐‘‹ is equal to the angle at ๐ด in polygon ๐ด๐ต๐ถ๐ท.

We already said that opposite angles in parallelograms are equal. So the angle at ๐ธ is also 70 degrees, which is equal to the angle at ๐ถ in polygon ๐ด๐ต๐ถ๐ท. We could use the same logic in parallelogram ๐ด๐ต๐ถ๐ท to show that the angles at ๐ต and ๐ท are each 110 degrees, which are the same as the angles at ๐น and ๐‘‹ in the larger polygon. Weโ€™ve shown then that all pairs of corresponding angles are indeed congruent. So our answer to the first check is yes.

Letโ€™s now consider whether corresponding pairs of sides are in proportion. Firstly, from the figure, we can see weโ€™ve been given side length ๐ด๐ต; itโ€™s 13 centimeters. And if the two polygons are similar, this will correspond to ๐บ๐น. We havenโ€™t been given the length of ๐บ๐น. But we know that opposite sides in a parallelogram are equal in length. So it will be the same as ๐‘‹๐ธ. Comparing the ratio of these two sides then, we find that ๐ด๐ต over ๐บ๐น is equal to 13 over 26, which simplifies to one-half.

The other pair of potentially corresponding sides weโ€™ve been given are ๐ต๐ถ and ๐ธ๐น. The ratio here is 11.5 over 23, which again simplifies to one-half. Now, Iโ€™ve written ๐ธ๐น here. But really, if weโ€™re to be consistent with the order of letters, then we should really write ๐น๐ธ as point ๐น corresponds to point ๐ต and point ๐ธ corresponds to point ๐ถ. However, in calculating the ratio, the length ๐ธ๐น is of course the same as the length ๐น๐ธ. So it makes no practical difference.

The ratio of these pairs of corresponding sides is therefore the same. As opposite sides in a parallelogram are equal in length, the same will be true for the remaining two pairs of corresponding sides. And so the answer to our second check, โ€œare corresponding pairs of sides in proportion?โ€, is also yes. Hence, both of the criteria for these two polygons to be similar are fulfilled. And so we can answer, yes, polygon ๐ด๐ต๐ถ๐ท is similar to polygon ๐บ๐น๐ธ๐‘‹.

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