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Geometry: High School15 chapters | 160 lessons

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Lesson Transcript

Instructor:
*Artem Cheprasov*

This lesson will teach you how to construct similar polygons, which are very useful in many quantitative fields. You'll learn that all of this has something to do with, of all things, snowflakes!

Imagine you're outside on a cold winter day with snow falling from the sky and you catch two identically shaped snowflakes, with one of them being twice as large as the other. Observe that while each edge of the larger snowflake is proportionally longer than its corresponding edge in the smaller snowflake. All of the corresponding angles are the same. This is just one real-life example of similar polygons. In case you're already lost with all the technical mumbo jumbo, let's define two important terms:

**Congruent**means equal, as in**congruent angles**, meaning angles that are equal to each other.- A
**polygon**is a closed, flat figure with three or more straight sides..

Here are some examples of polygons that should look familiar:

What makes one polygon similar to another? Two polygons are similar if and only if their vertices could be paired up, such that:

- The ratio of the lengths of corresponding sides is the same
- All corresponding interior angles are congruent

By **interior angles**, this means angles that are inside the polygon, as shown on the screen.

So how do we construct similar polygons?

You can construct similar polygons by following these steps:

- You need to have a reference polygon. Any polygon can serve as a reference as long as it is a 2-dimensional geometric shape with three or more straight sides.
- The to-be-constructed similar polygon has to follow the definition of what it means to be similar (namely, the ratio of all lengths of corresponding sides is the same, all corresponding interior angles are congruent). Decide what you want this ratio of corresponding sites to be. You're now ready to draw the new polygon.
- Using a ruler and a protractor, draw a point and two emanating line segments that have a corresponding vertex in the reference figure. The angle in the new figure should be the same as it corresponding angle in the reference polygon and the two line segments should be sized based on the ratio from step two.
- Each of your new line segments should lead to another vertex, as in the figure on the screen.

Repeat step three until the new similar polygon is complete.

As an example, consider the quadrilateral shown on the screen as our reference.

If we decide that the ratio of corresponding sides is equal to 2, the resulting similar polygon can be constructed as follows:

Start by duplicating the angle of 74 degrees at vertex *M* in our original figure. Since the line segments coming out of point *M* have lengths of 7.3 and 10, the line segments in the new figure should be twice as long: 14.6 and 20 respectively. We'll label the vertex corresponding to vertex *M* in the original figure as *Q* in the new figure. Next, we see in our image that the 7.3 units long line segment leads to vertex *L*, which measures 128 degrees. In the new figure, we should therefore have the 14.6 units long line segment leading to another vertex with an angle of 128 degrees. We label this vertex with the letter *P*. Continuing this process, we finally construct our similar polygon *QPSR* as shown on screen.

However, the orientation of a similar polygon does not have to be the same as that of the original. Let's take a closer look at what this actually means.

Similar polygons may be oriented any which way, as long as the ratio of the lengths of the corresponding sides remains the same and the corresponding angles remain congruent. In our previous example, the new polygon may be reflected over the *y*-axis, or over the *x*-axis, or rotated at some angle. The important thing to keep in mind is that the shape of the constructed figure has remained the same as per the definition.

Cool beans? Recall that a **polygon** is a closed, flat figure with three or more straight sides. Also, the term **congruent** means equal, as when we say congruent angles. Speaking of angles, remember that **interior angles** are the angles that are inside a polygon. Using these terms, we have to find two polygons to be similar if and only if their vertices could be paired up in such a way that:

- The ratio of the lengths of corresponding sides is the same.
- All corresponding interior angles are congruent.

Your preparation to construct similar polygons is now complete. So next time you're looking at the falling snowflakes on a cold winter day, take a moment and try to construct your own similar polygons. Fun, right?

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Geometry: High School15 chapters | 160 lessons

- Ratios and Proportions: Definition and Examples 5:17
- Geometric Mean: Definition and Formula 5:15
- Angle Bisector Theorem: Definition and Example 4:58
- Solving Problems Involving Proportions: Definition and Examples 5:22
- Similar Polygons: Definition and Examples 8:00
- The Transitive Property of Similar Triangles 4:50
- Triangle Proportionality Theorem 4:53
- Constructing Similar Polygons 4:59
- The Pythagorean Theorem: Practice and Application 7:33
- The Pythagorean Theorem: Converse and Special Cases 5:02
- Similar Triangles & the AA Criterion 5:07
- What is a Polygon? - Definition, Shapes & Angles 6:08
- Go to High School Geometry: Similar Polygons

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