What steps should you use to solve problems involving similar shapes? How do you know if two shapes are similar anyway? Well, we know that two shapes are said to be similar if they are the same shape but different size. An interesting fact is that corresponding angle measures within similar objects are congruent. Looking at angles between two shapes won't help us determine similarity at all. So we resort to looking at side lengths. Corresponding side lengths of similar shapes are proportional – all pairs can be expressed the same ratio.

When discussing similarity between two shapes, consider the following three rules. First please remember to investigate corresponding sides, and that corresponding means that we look at the same side relative to its position in the shape. Pay close attention to the pairings used in the upcoming examples. Second, use consistent notation. We learned in this module that a ratio can be expressed as two numbers written as a fraction, or separated by a colon. Whichever method you choose to use, you should also be sure to reduce the ratio to lowest terms, or its common ratio. Lastly, you must check all pairs of sides when verifying similarity. Just because one or two pairs of sides have a common ratio doesn't mean that all of them do. All pairs of side lengths must share the same common ratio in order for the shapes to be similar. Let's take a look at some examples now.

What is the similarity ratio, or common ratio, of triangle ABC to triangle DEF? Since triangle ABC was listed first, we should place side lengths from it first in our ratios. I'll go ahead and show both types of notation. Side AB corresponds to DE, and AC corresponds to DF, so we can write the ratios this way: ten over five and eight over four, or ten to five and eight to four. It is easy to see the fractions reduce to two over one, or two to one. This is our common ratio.

What is the common ratio for this pair of shapes? Pause the video now and check your answer in just a moment by resuming playback… Look at the three pairs of ratios; they all reduce to one over three, or one to three – our common ratio.

Next, consider this question; are these two shapes similar? Why or why not? Earlier in the video I mentioned that we must investigate all pairs of corresponding sides before we can answer the question. So the three ratios might look like this: thirty-five over seven, thirty over six, or twenty-four over five. We should now reduce each to a common ratio. The first two reduce nicely, but the third does not. In fact the fraction twenty-four over five does not reduce at all. So, even though the two objects appear to be the same shape, but different size, they are not actually similar because not all pairs of sides share the same common ratio.

Take a look at this next pair on your own and decide whether these two are similar. Be sure to explain why or why not… These are not similar. Even though all ratio pairs reduce, they do not reduce to the same common ratio. Side AB and side CD produce a strange ratio.

Now, let's take this one step further. Looking at these two shapes, knowing that they are similar, what should we do to find the missing side lengths, x and y? Since the side length ratios are the same in similar figures, we would only need to find it once and apply the ratio to find the missing sides. It appears the common ratio from the larger quadrilateral to the smaller is two to one – the side lengths on the left are twice as large as those on the right, the side lengths on the right are half of those on the left. So x must be four and y must be 1.

Try this last real-world example on your own. We told that triangles formed between two objects and their shadows are similar in shape. You can find a common ratio and use that to determine the missing side length – the height of the Statue of Liberty. Pause the video now, and check your answer by resuming playback in just a moment… It appears the common ratio is twelve to one, so twenty-five times twelve gives us the height of the statue to be three hundred feet.