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Glencoe Geometry: Online Textbook Help13 chapters | 152 lessons

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Lesson Transcript

Instructor:
*Elizabeth Foster*

Elizabeth has been involved with tutoring since high school and has a B.A. in Classics.

Similar polygons have the same angles and proportional side lengths. In this lesson, you'll solve some practice problems using their special properties.

What do we mean when we say that we have **similar polygons**? It means that corresponding angles have the same measures, and corresponding sides have proportional lengths.

For example, in this drawing, you can see that the corresponding angles of both polygons, labeled *a* through *e*, are all the same.

Each side of the bigger polygon is twice as long as the corresponding side of the smaller polygon, so the sides are proportional in a ratio of 2:1. That means these two polygons are similar. The perimeters of the polygons will also be proportional in the same ratio.

If you know that two polygons are similar, you can use the side lengths and angle measures of one polygon to help you figure out the side lengths and angle measures of the other. In this lesson, we'll look at some practice problems to give you an idea.

Polygon *A* is similar to Polygon *B*. The perimeter of Polygon *A* is 15 meters. The perimeter of Polygon *B* is 10 meters. *x* = *w*, and *y* = *v*. Given that the length of side *w*1 is 3 meters, what is the length of side *x*?

If you don't know what to do with a problem like this, a good first step is to mark up the diagram with all the information given. The problem tells us that the perimeter of *A* is 15 meters, and the perimeter of *B* is 10 meters. If we simplify that, we get a ratio of 1.5 to 1, which means that every side of Polygon *A* is 1.5 times as long as the corresponding side of Polygon *B*. We'll use this to simplify the diagram. We know that *w*1 is equal to 1.5 times *w*, so instead of *w*1, we'll write 1.5*w*. The same goes for all the sides of Polygon *A*, so we'll replace them all.

The problem also tells us that *w*1, which we just renamed as 1.5*w*, is equal to 3. If 1.5*w* = 3, then *w* must equal 3/1.5, meaning that *w* = 2. The problem tells us that *x* = *w*, so *x* must also be 2.

Ready for something a little trickier? You know you got this. The two arrows shown are congruent polygons.

The measure of angle *a* is 60 degrees, and the measure of angle *b* is also 60 degrees. The length of side *x* is 1/3 the length of side *y*. The length of side *y* is 12 inches. What's the area of the region shown in green?

All right, let's start by writing everything we know on the diagram. We know that angles *A* and *B* are both 60 degrees, so we'll start by marking that on both shapes. The problem tells us that *y* = 12, so we'll write that on the diagram too. The length of *x* is 1/3 the length of *y*, so side *x* is 4.

Now look at the 'point' of each arrow as a triangle.

Part of one side of the triangle is cut out, but it would be right where the orange line is in the picture. In each triangle, two of the angles are 60 degrees. That means the third angle also has to be 60 degrees, because all the angles in a triangle add up to 180. In other words, the point of each arrow is basically an equilateral triangle with a chunk cut out of one side.

In an equilateral triangle, all sides are the same and all angles are the same. So, we'll mark that on the diagram. We know that each side of the big triangle is three times the length of the side of the smaller triangle, so we can label the sides of the small triangle as *x*, and the sides of the big triangle as 3*x*. Okay, now we're talking! We know that *x* = 4, so we can replace *x* and 3*x* with 4 and 12, respectively.

Now we just need to find the area of the big triangle and subtract the area of the smaller triangle. We'll first find the height of the big triangle, using the Pythagorean Theorem. The Pythagorean Theorem states that *a*^2 + *b*^2 = *c*^2, where *a* and *b* are the two shorter sides of a triangle, and *c* is the longest side.

Using the Pythagorean Theorem, we find that the height of the triangle is 10.4 square inches. The area of a triangle is 1/2 times the base times the height, or 1/2 * 12 * 10.4, so the area of the big triangle is roughly 62.4 square inches.

We don't need to repeat the Pythagorean Theorem for the smaller triangle, because we know that the altitudes of similar triangles are proportional, so the height of the small triangle will be 1/3 the height of the large triangle, which is roughly 3.47 inches. That makes the area of the small triangle equal to 6.93 square inches. Note that the areas of the two triangles are not proportional! The total area of the green shading is 62.4 - 6.93, or 55.4 square inches.

In this lesson, you worked through two practice problems with similar polygons. **Similar polygons** have the same corresponding angles and proportional corresponding sides.

To do problems with similar polygons, start by writing everything you know on the diagram. Then figure out the proportion between the sides of the larger and the smaller polygon, and use it to:

- Find as many side lengths as you can.
- Label the side lengths of one polygon in terms of the other. For example, if the smaller shape has a side length
*x*, the larger shape might have a side labeled 2*x*or 3*x*.

Then use the proportions you know to solve for the side lengths you don't know.

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Glencoe Geometry: Online Textbook Help13 chapters | 152 lessons

- Ratios and Proportions: Definition and Examples 5:17
- Solving Problems Involving Proportions: Definition and Examples 5:22
- Similar Polygons: Definition and Examples 8:00
- Similar Polygons: Practice Problems 5:30
- Applications of Similar Triangles 6:23
- The Transitive Property of Similar Triangles 4:50
- Triangle Proportionality Theorem 4:53
- Midsegment: Theorem & Formula 4:18
- Proportional Relationships in Triangles 6:28
- Angle Bisector Theorem: Proof and Example 6:12
- Fractals in Math: Definition & Description 5:36
- Go to Glencoe Geometry Chapter 6: Proportions and Similarity

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