Back To CourseGeometry: High School
15 chapters | 160 lessons
As a member, you'll also get unlimited access to over 75,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.Try it risk-free
Amy has a master's degree in secondary education and has taught math at a public charter high school.
Imagine you're a famous artist. You've been contracted by your city to build two sets of similar figures in front of the city hall for a modern art installation. The city wants to have a pair of similar triangles and a pair of similar rectangles. In math, we say that two figures are similar when the shapes are the same with the only difference being the size.
For example, a model car is similar to the real life car that it models. A model house is similar to the real life house that it models. If the model is built to scale, it has the same proportions as the real life object; the only way you can tell them apart is that they are different sizes. Even if one object was rotated a different way, you could still see that they look the same except for the size.
As the artist for the city now, you need to recognize when two figures are similar. You're looking at a pair of triangles and a pair of rectangles that you've made. What are the criteria for checking to see if two figures are similar? There are two.
The first is that the corresponding angles are all congruent to each other. Corresponding means they are located at the same location if the two figures are placed in the same way. For example, the right angles of two right triangles are corresponding angles. Even if you rotated one triangle, that right angle is still the corresponding angle to the other right angle.
The second criterion is that all the corresponding sides are proportional to each other. This means that if one pair of corresponding sides has a ratio of 2:1, meaning that one figure is twice as large as the other, then all the other corresponding sides will have the same ratio. Always remember to simplify these ratios as you would fractions.
You can see that it's not hard to spot corresponding angles and corresponding sides when the figures are placed in the same way, but what if the figures are arranged differently? For example, it may be a little more difficult to identify corresponding parts between two triangles when one triangle is rotated differently, flipped upside down, or even just far away from the other triangle.
Luckily, we have transformations, or ways to rearrange a figure. There are several transformations to help us place our figures in the same way. You can use rotation to turn one figure until it's placed in the same way as the other figure. If one of your figures were upside down, then you can use a reflection, or flip it across a line, to get it right side up. If one figure is far away from the other, perform a translation, or slide the figure until it's closer. All three of these transformations won't change the size or shape of the figure, so you'll still have all of your corresponding angles and corresponding sides if the two figures are similar.
Now that you know how to use transformations to place your figures in the same way, let's see how you can figure out if your two triangles and two rectangles are similar.
The pair of triangles that you are looking at are these two.
No transformations (no rotations, no reflections, and no translations) are needed here, as both triangles are facing the same way. You can easily see the corresponding angles and sides. The top angles are corresponding angles. The bottom left angles are corresponding and so are the bottom right angles of each triangle. The bottom sides of each triangle are corresponding. The left straight edge sides are corresponding. And, the slanted edges are corresponding.
In order for these two triangles to be similar, all the corresponding angles have to be congruent. They have to be the same. All the corresponding sides must be proportional. You look at the corresponding angles of your two triangles first. Are they are all the same? Yes, they are. You have 90 and 90, 37 and 37, and 53 and 53. This part checks off.
Next, you look at your corresponding sides. Are they are all proportional by the same amount? Let's see. Starting with the bottom side and working your way clockwise, you get 6:3, which simplifies to 2:1. The next side, the straight side on the left, gives you a ratio of 8:4, which simplifies to 2:1. Last, the slanted edge gives you a ratio of 10:5, which also simplifies to 2:1. Keeping your pattern of larger triangle to smaller triangle, you see that the ratios for each set of corresponding sides are all the same. This means that your two triangles are similar because they have congruent corresponding angles and proportional corresponding sides.
Now, let's check the two rectangles. Are these similar?
You need to check two things. You need to make sure that all corresponding angles are congruent and that all corresponding sides are proportional. Oh, but look, your two shapes are not facing the same way. Because they are not facing the same way, you can't say that the bottom sides are corresponding. You have to make sure that the bottom side of each figure really is the bottom side if the figures are both facing the same way. You need to perform a transformation on one of the rectangles so they both are facing the same way. You can rotate the smaller rectangle so that it is laying flat like the larger rectangle. And then, you can translate the smaller rectangle so it is directly under the bigger rectangle. This way, you can easily see what the corresponding angles and corresponding sides are. You get this.
Now you can easily see which angles are corresponding and which sides are corresponding. You know that since these are two rectangles, all the angles are 90 degrees. Since both shapes are rectangles, then all the corresponding angles are congruent. They are all 90 degrees.
Now, the sides. Because the rectangle has two pairs of congruent and parallel lines, you only need to compare those two different sides. You need to compare the width size and the length size. Looking at your rectangles, you see that the length of the larger rectangle is 12 and length of the smaller rectangle is 4. The larger to smaller ratio is 12:4. Simplifying this gives you 3:1. Next, you need to compare the width sizes. The larger rectangle has a width of 6 while the smaller rectangle has a width of 2. The ratio here is 6:2 which simplifies to 3:1. Are these ratios the same? Yes, they are. So, this tells you that the two rectangles are also similar. Yay! Your art installation is a success. Both pairs of shapes are similar.
When doing these similarity problems, always perform the ratio test for all corresponding sides and the congruence test for all corresponding angles, even if the shapes look identical. Sometimes, you may find that two similar looking shapes aren't similar at all.
Let's review what you've learned. We say that two figures are similar when the shapes are the same with the only difference being the size. To check whether two figures are similar, you need to check two things. First, you need to check to see if all the corresponding angles are congruent. Second, you need to check to see if all the corresponding sides are proportional. The proportion needs to be the same for all corresponding sides.
It's easy to locate corresponding angles and corresponding sides when similar figures are placed in the same way. If two figures are arranged differently, you can use several transformations, or ways to rearrange a figure. You can perform a rotation or turn the figure, like we did with the rectangles. You can do a reflection, or flip the figure across a line. A translation simply lets you slide a figure without changing its orientation. These three transformations won't change the size or shape of the figure. You'll still have all the corresponding parts and should be able to match them together with less difficulty.
To unlock this lesson you must be a Study.com Member.
Create your account
Already a member? Log InBack
Did you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.
To learn more, visit our Earning Credit Page
Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.
Back To CourseGeometry: High School
15 chapters | 160 lessons