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Similar Polygons and Scale Factors

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Similar Polygons and Scale Factors

What if you were told that two pentagons were similar and you were given the lengths of each pentagon's sides. How could you determine the scale factor of pentagon #1 to pentagon #2? After completing this Concept, you'll be able to answer questions like this one about similar polygons.

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CK-12 Similar Polygons and Scale Factors

James Sousa: Similar Polygons

Now watch this video.

James Sousa: Scale Factor

Guidance

Similar polygons are two polygons with the same shape, but not the same size. Similar polygons have corresponding angles that are congruent, and corresponding sides that are proportional.

These polygons are not similar:

Think about similar polygons as enlarging or shrinking the same shape. The symbol \sim is used to represent similarity. Specific types of triangles, quadrilaterals, and polygons will always be similar. For example, all equilateral triangles are similar and all squares are similar. If two polygons are similar, we know the lengths of corresponding sides are proportional. In similar polygons, the ratio of one side of a polygon to the corresponding side of the other is called the scale factor. The ratio of all parts of a polygon (including the perimeters, diagonals, medians, midsegments, altitudes) is the same as the ratio of the sides.

Example A

Suppose \triangle ABC \sim \triangle JKL. Based on the similarity statement, which angles are congruent and which sides are proportional?

Just like in a congruence statement, the congruent angles line up within the similarity statement. So, \angle A \cong \angle J, \angle B \cong \angle K, and \angle C \cong \angle L. Write the sides in a proportion: \frac{AB}{JK} = \frac{BC}{KL} = \frac{AC}{JL}. Note that the proportion could be written in different ways. For example, \frac{AB}{BC} = \frac{JK}{KL} is also true.

Example B

MNPQ \sim RSTU. What are the values of x, y and z?

In the similarity statement, \angle M \cong \angle R, so z = 115^{\circ}. For x and y, set up proportions.

\frac{18}{30} &= \frac{x}{25} && \ \frac{18}{30} = \frac{15}{y}\\450 &= 30x && 18y = 450\\x &= 15 && \quad y = 25

Example C

ABCD \sim AMNP. Find the scale factor and the length of BC.

Line up the corresponding sides, AB and AM = CD, so the scale factor is \frac{30}{45} = \frac{2}{3} or \frac{3}{2}. Because BC is in the bigger rectangle, we will multiply 40 by \frac{3}{2} because \frac{3}{2} is greater than 1. BC = \frac{3}{2} (40)=60.

CK-12 Similar Polygons and Scale Factors

Guided Practice

1. ABCD and UVWX are below. Are these two rectangles similar?

2. What is the scale factor of \triangle ABC to \triangle XYZ? Write the similarity statement.

3. \triangle ABC \sim \triangle MNP. The perimeter of \triangle ABC is 150, AB = 32 and MN = 48. Find the perimeter of \triangle MNP.

Answers:

1. All the corresponding angles are congruent because the shapes are rectangles.

Let’s see if the sides are proportional. \frac{8}{12} = \frac{2}{3} and \frac{18}{24} = \frac{3}{4}. \frac{2}{3} \neq  \frac{3}{4}, so the sides are not in the same proportion, and the rectangles are not similar.

2. All the sides are in the same ratio. Pick the two largest (or smallest) sides to find the ratio.

\frac{15}{20} = \frac{3}{4}

For the similarity statement, line up the proportional sides. AB \rightarrow XY, BC \rightarrow XZ, AC \rightarrow YZ, so \triangle ABC \sim \triangle YXZ.

3. From the similarity statement, AB and MN are corresponding sides. The scale factor is \frac{32}{48} = \frac{2}{3} or  \frac{3}{2}. \triangle ABC is the smaller triangle, so the perimeter of \triangle MNP is \frac{3}{2} (150)=225.

Practice

For questions 1-8, determine whether the following statements are true or false.

  1. All equilateral triangles are similar.
  2. All isosceles triangles are similar.
  3. All rectangles are similar.
  4. All rhombuses are similar.
  5. All squares are similar.
  6. All congruent polygons are similar.
  7. All similar polygons are congruent.
  8. All regular pentagons are similar.
  9. \triangle BIG \sim \triangle HAT. List the congruent angles and proportions for the sides.
  10. If BI = 9 and HA = 15, find the scale factor.
  11. If BG = 21, find HT.
  12. If AT = 45, find IG.
  13. Find the perimeter of \triangle BIG and \triangle HAT. What is the ratio of the perimeters?
  14. An NBA basketball court is a rectangle that is 94 feet by 50 feet. A high school basketball court is a rectangle that is 84 feet by 50 feet. Are the two rectangles similar?
  15. HD TVs have sides in a ratio of 16:9. Non-HD TVs have sides in a ratio of 4:3. Are these two ratios equivalent?

Use the picture to the right to answer questions 16-20.

  1. Find m \angle E and m \angle Q.
  2. ABCDE \sim QLMNP, find the scale factor.
  3. Find BC.
  4. Find CD.
  5. Find NP.

Determine if the following triangles and quadrilaterals are similar. If they are, write the similarity statement.

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