Back To CourseGeometry: High School
15 chapters | 160 lessons
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Amy has a master's degree in secondary education and has taught math at a public charter high school.
To begin, let's talk about polygons and what they are. Polygons are defined as two-dimensional shapes with straight sides. If you take a pen or pencil starting at one point on your piece of paper and, without lifting your pen or pencil, you draw straight lines until you return to your starting point, you will have drawn a polygon. Our most common shapes, such as triangles, squares, and rectangles, are all polygons. Circles are not, because they have curved sides. A star is a polygon because all its sides are straight.
When we say that two polygons are similar, we mean that the only difference between them is size. If one is the exact same shape as the other, just smaller or larger, then they are similar.
Sometimes, one of the shapes may be flipped around or rotated, but the shapes are still similar. It's like you cutting two similar triangles out of a piece of paper. You rotate and flip one of them. Now the shapes look different at first glance, but you know that they are still similar polygons. When working with problems, keep this point in mind. Look at the two shapes and imagine flipping and rotating them to see if they really are different or if they are similar.
Now, imagine taking a shape and stretching it out to make it bigger. What do you notice about the shape? You notice that it keeps it proportions. You also see that you can place your original shape inside the larger shape and match each of the angles together. You see that the only difference is the length of the sides.
What you have just found is that the one distinguishing aspect of similar polygons is that all the angles remain the same. No matter how much you increase or decrease the size of the shapes, the angles will always remain the same measurement. If one angle in a triangle measures 60 degrees, then the same angle in a similar triangle will also measure 60 degrees.
Let's try comparing two triangles together to see if they are similar. Look at these triangles. When you first look at these two triangles, what do you see? Can you tell that they are similar at first glance? What I do is I imagine rotating one of them in my head to see what happens, to see if that shape starts to resemble the other shape. I pick the smaller one to rotate. I see that if I rotate the shape 180 degrees, or halfway around a circle, that the smaller one looks a lot like the bigger one. I remember that the one distinguishing mark for similar polygons is that all the matching angles must equal each other, so I look at my angles to see if they are the same. I see that my two little angles are the same and my larger angle is also the same. What do you know? These are similar triangles!
Some problems you encounter might ask you to find the angle of one shape given that the shape is similar to another shape that has the angle marked. In this case, you would rotate or flip the shape until they look similar, then you can match the angles to each other to find the angle measurement. For example, say you were given two triangles like these.
You are told that the triangles are similar and the problem is asking you to find the measure of angle A. What do you do? First, you look to see if you have to rotate or flip. You see that the two triangles already look alike, so now all you have to do is match the angles. Angle A is the top angle in the smaller triangle, so what is the top angle in the larger triangle? You see that it is 75, so the answer is 75 degrees.
Because the only difference between two similar polygons is the size, the two shapes will always be proportional to each other. This is why you can say that one shape is twice as big as the other. What you are saying is that each side of the larger shape is 2 times the length of each matching side in the smaller shape.
Look at these two similar shapes. I can say that the larger rectangle is two times bigger than the smaller rectangle. Why? Because if I took the measurement of each side of the larger rectangle and divided it by the measurement of the matching side of the smaller rectangle, my answer will always be 2. Take the short sides that measure 4 in the larger and 2 in the smaller. What is 4 divided 2? It's 2. What about the longer sides? What is 8 divided by 4? That's 2 as well. So my larger rectangle is twice as big.
We can use proportions to help us find missing sides as well. Say in our rectangle problem we wanted to find the measurement of the longer side of the small rectangle. Say that we didn't know it measured 4. We are told that the rectangles are similar. But what do we do? Because the short side of the larger rectangle measures 4 while the matching short side of the smaller rectangle measures 2, we can divide 4 by 2 to get a proportion of 2. Because all the sides of similar polygons must be the same proportion, my 8 divided by my unknown side must also equal 2. I can write an algebraic expression for this using the variable x for the side that I want to find.
4/2 = 8/x
2 = 8/x
Now, with some algebraic manipulation, I can solve for x to find my answer.
2 = 8/x
2*x = (8/x) * x
2x = 8
2x/2 = 8/2
x = 4
As expected, my answer is 4.
We've learned that polygons are two-dimensional shapes with straight sides. When two polygons are similar it means that the only difference between them is their size. When you resize any shape, the shape keeps its proportions and maintains all the angle measurements. Matching angles in similar polygons will always measure the same. When two shapes are proportionate, you know that if you divide matching sides, your answer will be the same for all sides. This is why you can say that one shape is so many times bigger than another. If two triangles are similar to each other, then the sides are proportionate. If the sides are proportionate, then if the larger triangle is 2 times bigger, your answer when you divide a side of the larger triangle with the matching side of the smaller triangle will be 2.
After watching this lesson, you should be able to interpret how two polygons are similar to each other by comparing angles and proportions.
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Back To CourseGeometry: High School
15 chapters | 160 lessons