In this video, we will learn how to use the properties of similar polygons to find unknown angles, side lengths, scale factors, and perimeters. Firstly, we recall that a polygon is a closed shape with straight sides. So, for example, a triangle, a square, and a hexagon are all examples of polygons, whereas a circle isn’t as its sides are not all straight. Two polygons are said to be mathematically similar if one is an enlargement of the other. This means that two key things will be true.
Firstly, corresponding angles in the two similar polygons are congruent or identical. Secondly, corresponding pairs of sides are in proportion. Another way of saying this is to say that corresponding pairs of sides are all in the same ratio. So in our example here, the length of 𝐴𝐵 divided by the length of the corresponding side 𝐸𝐹 gives the same result as if we divide 𝐵𝐶 by 𝐹𝐺, and if we divide 𝐶𝐷 by 𝐺𝐻, and if we divide 𝐷𝐴 by 𝐻𝐸.
If two polygons are similar, then we can express this using the notation on screen. Notice that the order of the letters is important. So if I’ve said 𝐴𝐵𝐶𝐷 is similar to 𝐸𝐹𝐺𝐻, then this means that the angle at 𝐴 corresponds to the angle at 𝐸. And the side 𝐶𝐷 corresponds to the side 𝐺𝐻. If two polygons are similar, then we can calculate the scale factor between them. This is the ratio of corresponding lengths and will always be a multiplier. It’s the value we need to multiply the length in one polygon by to find the corresponding length in the other polygon. We can calculate the scale factor in either direction.
For example, if we need to multiply the lengths of 𝐴𝐵𝐶𝐷 by 𝑆 in order to give the corresponding lengths in 𝐸𝐹𝐺𝐻, then in order to go the other way, we’d need to multiply the lengths of 𝐸𝐹𝐺𝐻 by one over 𝑆 to give the corresponding lengths in 𝐴𝐵𝐶𝐷. It’s important to notice that the scale factor is always a multiplier. So, in formulae, we may think of going from the larger polygon to the smaller polygon as dividing by 𝑆. But our scale factor will be the multiplier one over 𝑆. Let’s now look at a question where we need to calculate the scale factor between two similar polygons and then determine the length of a side.
Given that the rectangles shown are similar, what is 𝑥?
We’re told in the question that these two rectangles are mathematically similar, which means that two things are true. Firstly, all pairs of corresponding angles between the two rectangles are congruent. Now, for a pair of rectangles, this is true even for nonsimilar rectangles as all the interior angles in a rectangle are 90 degrees. Secondly, and more importantly here, all pairs of corresponding sides are in proportion.
We’re looking to find the value of 𝑥 which represents a side length in the larger rectangle. We therefore need to know the scale factor 𝑆 that is the multiplier that takes us from the smaller rectangle to the larger. We can use any pair of corresponding sides to work out the scale factor. It’s equal to the new length divided by the original length. From the figure, we can see that we have a pair of corresponding sides of 29 centimeters on the smaller rectangle and 58 centimeters on the larger. We therefore divide the new length of 58 by the original length of 29, giving a scale factor of 58 over 29, which simplifies to two.
Remember, the scale factor is always a multiplier. So this tells us that the lengths on the larger rectangle are all twice the corresponding lengths on the smaller rectangle. To work out the value of 𝑥 then, we need to take the corresponding length on the smaller rectangle, 26, and multiply it by the scale factor of two, which gives 52. So we found the value of 𝑥. If we had wanted to calculate a length on the smaller rectangle rather than one on the larger rectangle, we could’ve used the scale factor of one-half. That’s the reciprocal of two. This could’ve been found by dividing the length of 29 centimeters by the corresponding length of 58 centimeters.
We can check our value for 𝑥 by multiplying it by one-half, so 52 multiplied by one-half or half of 52, which is indeed equal to 26. In formulae, we may think of this as dividing by two. But remember, the scale factor is always a multiplier, so we’re using a multiplier of one-half.
Let’s consider another example.
If the two following polygons are similar, find the value of 𝑥.
We’re told that these two polygons are mathematically similar, which means that two key things are true. Firstly, corresponding pairs of angles are congruent. And secondly, corresponding pairs of sides are in proportion or in the same ratio. We want to calculate this ratio or scale factor. And in this case, as 𝑥 represents the length on the smaller polygon, we’re working in this direction. The scale factor can be calculated from any corresponding pair of sides by dividing the new length by the original length.
We have a corresponding pair of side lengths of 34 centimeters and 85 centimeters. So dividing the new length, that’s the length which is in the same polygon as the one we want to calculate, by the original length, we have a scale factor of 34 over 85. This can be simplified by dividing both the numerator and denominator by 17 to give two-fifths. Notice that this value feels right. We’re moving from the larger polygon to the smaller one. So the length should be getting smaller.
Our scale factor is a fractional value less than one. And when we multiply something by a fractional value less than one, such as two-fifths, it will become smaller. To calculate the value of 𝑥 then, we need to take the corresponding length on the larger polygon, which is 75 centimeters, and multiply it by our scale factor of two-fifths. We can simplify this calculation by canceling a factor of five from both the numerator and denominator to give 𝑥 equals 15 multiplied by two, which is of course 30. So we’ve found the value of 𝑥.
We can check our answer by confirming that the ratio of corresponding pair of sides is indeed the same. We have 30 over 75 and 34 over 85. Both fractions do indeed cancel to our scale factor of two-fifths. And so this confirms that our answer is correct.
In our next example, we’ll see how we can test whether or not two polygons are similar.
Is polygon 𝐴𝐵𝐶𝐷 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 𝐵 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 𝐴𝐵𝐶𝐷, 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 𝐴𝐵𝐶𝐷, the angle at 𝐶 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 𝐴𝐵𝐶𝐷.
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 𝐶 in polygon 𝐴𝐵𝐶𝐷. We could use the same logic in parallelogram 𝐴𝐵𝐶𝐷 to show that the angles at 𝐵 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 𝐴𝐵; 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 𝐴𝐵 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 𝐵𝐶 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 𝐵 and point 𝐸 corresponds to point 𝐶. 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, “are 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 𝐴𝐵𝐶𝐷 is similar to polygon 𝐺𝐹𝐸𝑋.
Let’s now consider another example in which we prove similarity of two polygons.
Are these two polygons similar? If yes, find the scale factor from 𝑋𝑌𝑍𝐿 to 𝐴𝐵𝐶𝐷.
To determine whether these two polygons are similar, we need to consider two questions. Firstly, are corresponding pairs of angles congruent? And secondly, are corresponding pairs of sides in proportion? Remember that the order of the letters is important. So if these two polygons are similar, then the angle at 𝑋 will correspond with the angle at 𝐴. And, for example, the side 𝑍𝐿 will correspond with the side 𝐶𝐷.
Let’s consider the angles first. And looking at the figure, we can see that three of the angles in each polygon have been marked using different notation. The single line marking the angle at 𝐿 and the angle at 𝐷 indicates that these two angles are equal. The pair of lines at angle 𝑋 and angle 𝐴 indicates that these two angles are equal. In the same way, the triple line at angle 𝑍 and angle 𝐶 indicates that these two angles are equal. But what about the final angle in each polygon? Well, each of these polygons have four sides. They are quadrilaterals. And we know that the sum of the interior angles in any quadrilateral is 360 degrees.
If all of the other three angles in the two quadrilaterals are the same, then in each case, we will have the same amount left over from the total of 360 degrees to form the final angle. And therefore, we can also say that angle 𝑌 is equal to angle 𝐵. So we’ve seen that all pairs of corresponding angles are indeed congruent between the two polygons. So the answer to our first check is yes.
Next, let’s consider the sides. And we need to check whether corresponding pairs of sides are in proportion. That is, do we get the same value when we divide each side of one polygon by the corresponding side of the other? We’ve been given all of the side lengths we need. So we can substitute them and work out what each of these ratios simplify to. In fact, they are all equal to four-fifths or 0.8. And so corresponding pairs of sides are indeed in proportion. As we’ve answered yes to both statements, we can conclude that the two polygons are indeed similar.
We’re now asked to find the scale factor from 𝑋𝑌𝑍𝐿 to 𝐴𝐵𝐶𝐷. That’s traveling in this direction. Remember, the scale factor is the value we multiply the lengths on the first polygon by in order to give the corresponding lengths on the second. And in fact, we’ve already worked this out. It’s the value we get when we divide a new length, that’s a length on the polygon we’re going to, by an original length. That’s the corresponding length on the polygon we’re coming from.
We’ve already seen that when we divide each length on 𝐴𝐵𝐶𝐷 by the corresponding length on 𝑋𝑌𝑍𝐿, we get the value four-fifths. So this is our scale factor. Notice that this value does make logical sense. The lengths on 𝐴𝐵𝐶𝐷 are smaller than the lengths on 𝑋𝑌𝑍𝐿. So our scale factor should be a fractional value less than one. So we’ve completed the problem. We found that these two polygons are similar. And the scale factor going from 𝑋𝑌𝑍𝐿 to 𝐴𝐵𝐶𝐷, that’s from the larger polygon to the smaller, as a decimal is 0.8.
Just as a reminder, if we were going back the other way, so finding the scale factor from 𝐴𝐵𝐶𝐷 to 𝑋𝑌𝑍𝐿, then we would have the reciprocal. The reciprocal of a fraction can be found by flipping it, so swapping its numerator and denominator. The scale factor going in the other direction would be the multiplier of five over four. Again, this makes sense. Five over four is a little bit bigger than one. It’s 1.25 as a decimal. And so we’re multiplying the lengths on 𝐴𝐵𝐶𝐷 by something bigger than one, which will give larger values for the corresponding lengths on 𝑋𝑌𝑍𝐿.
Let’s now summarize some of the key points from this video. Firstly, two polygons are similar if corresponding pairs of angles are congruent and all corresponding pairs of sides are in proportion. To calculate the scale factor between two similar polygons, we can use the lengths of any pair of corresponding sides. And we divide the new length, that’s the length on the polygon we’re going to, by the corresponding length on the original. That’s the polygon we’re coming from.
Remember that the scale factor is always a multiplier. When we’re going from the smaller shape to the larger shape, we will always have a scale factor greater than one. When going in the other direction, so going from the larger shape to the smaller shape, the scale factor will be the reciprocal of the scale factor working in the other direction. It will be one over 𝑆. And in this case, the scale factor will be less than one. In formulae, we may think of this as dividing by the previous scale factor of 𝑆. But our scale factor should really be a multiplier.
Once we know that two polygons are similar, we can calculate the measures of any missing angles. And we can use scale factors to calculate the lengths of any missing sides, provided we’ve been given the corresponding information on the other polygon.