Video: Verifying Whether the Two Given Polygons Are Similar

Is polygon 𝐴𝐡𝐢𝐷 similar to polygon 𝐺𝐹𝐸𝑋?

04:24

Video Transcript

Is polygon 𝐴𝐡𝐢𝐷 similar to polygon 𝐺𝐹𝐸𝑋?

From the figure, we can see that the two polygons we’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 they’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, it’s important to remember that when we’re working with similar polygons, the order of the letters is important. So if these polygons are similar, then the angle at 𝐴 will correspond to the angle at 𝐺. The angle at 𝐡 will correspond to the angle at 𝐹, and so on. We can therefore deduce that the polygons have been drawn in the same orientation.

Let’s consider the angles first of all then. In polygon 𝐺𝐹𝐸𝑋, we’ve been given a marked angle of 110 degrees. And in polygon 𝐴𝐡𝐢𝐷, we’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 𝐺𝐹𝐸𝑋, the angle at 𝑋 will be 110 degrees. And in parallelogram 𝐴𝐡𝐢𝐷, the angle at 𝐢 will be 70 degrees.

Let’s consider the angle at 𝐺 in the polygon 𝐺𝐹𝐸𝑋. By extending the line 𝐹𝐺, we now see that we have two parallel lines 𝑋𝐺 and 𝐸𝐹 and a transversal 𝐺𝐹. We know that corresponding angles in parallel lines are equal, which means that the angle above the line 𝑋𝐺 will be equal to the angle above the line 𝐸𝐹. It’s 110 degrees. We also know that angles on a straight line sum to 180 degrees, which means the angle below 𝑋𝐺 will be 180 minus 110. It’s 70 degrees. This shows us that the angle at 𝐺 in polygon 𝐺𝐹𝐸𝑋 is equal to the angle at 𝐴 in polygon 𝐴𝐡𝐢𝐷.

We already said that opposite angles in parallelograms are equal. So the angle at 𝐸 is also 70 degrees, which is equal to the angle at 𝐢 in polygon 𝐴𝐡𝐢𝐷. We could use the same logic in parallelogram 𝐴𝐡𝐢𝐷 to show that the angles at 𝐡 and 𝐷 are each 110 degrees, which are the same as the angles at 𝐹 and 𝑋 in the larger polygon. We’ve shown then that all pairs of corresponding angles are indeed congruent. So our answer to the first check is yes.

Let’s now consider whether corresponding pairs of sides are in proportion. Firstly, from the figure, we can see we’ve been given side length 𝐴𝐡; it’s 13 centimeters. And if the two polygons are similar, this will correspond to 𝐺𝐹. We haven’t been given the length of 𝐺𝐹. But we know that opposite sides in a parallelogram are equal in length. So it will be the same as 𝑋𝐸. Comparing the ratio of these two sides then, we find that 𝐴𝐡 over 𝐺𝐹 is equal to 13 over 26, which simplifies to one-half.

The other pair of potentially corresponding sides we’ve been given are 𝐡𝐢 and 𝐸𝐹. The ratio here is 11.5 over 23, which again simplifies to one-half. Now, I’ve written 𝐸𝐹 here. But really, if we’re to be consistent with the order of letters, then we should really write 𝐹𝐸 as point 𝐹 corresponds to point 𝐡 and point 𝐸 corresponds to point 𝐢. However, in calculating the ratio, the length 𝐸𝐹 is of course the same as the length 𝐹𝐸. 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, β€œare corresponding pairs of sides in proportion?”, is also yes. Hence, both of the criteria for these two polygons to be similar are fulfilled. And so we can answer, yes, polygon 𝐴𝐡𝐢𝐷 is similar to polygon 𝐺𝐹𝐸𝑋.

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