- Recognize similar polygons.
- Identify corresponding angles and sides of similar polygons from a similarity statement.
- Use scale factors.
- Solve the proportions.
- In the picture, ABXZ=BCXY=ACYZ.
- Find AB.
- Find BC.
- What is AB:XZ?
Know What? A baseball diamond is a square with 90 foot sides. A softball diamond is a square with 60 foot sides. Are the two diamonds similar? If so, what is the scale factor?
Think about similar polygons as enlarging or shrinking the same shape. The symbol ∼ is used to represent similar.
Similar Polygons: Two polygons with the same shape, but not the same size. The corresponding angles are congruent, and the corresponding sides are proportional.
These polygons are not similar:
Example 1: Suppose △ABC∼△JKL. Based on the similarity statement, which angles are congruent and which sides are proportional?
Solution: Just like a congruence statement, the congruent angles line up within the statement. So, ∠A≅∠J,∠B≅∠K, and ∠C≅∠L. Write the sides in a proportion, ABJK=BCKL=ACJL.
Because of the corollaries we learned in the last section, the proportions in Example 1 could be written several different ways. For example, ABBC=JKKL is also true.
Example 2: MNPQ∼RSTU. What are the values of x,y and z?
Solution: In the similarity statement, ∠M≅∠R, so z=115∘. For x and y, set up a proportion.
Specific types of triangles, quadrilaterals, and polygons will always be similar. For example, all equilateral triangles are similar and all squares are similar.
Example 3: ABCD and UVWX are below. Are these two rectangles similar?
Solution: All the corresponding angles are congruent because the shapes are rectangles.
Let’s see if the sides are proportional. 812=23 and 1824=34. 23≠34, so the sides are not in the same proportion, so the rectangles are not similar.
If two polygons are similar, we know the lengths of corresponding sides are proportional.
Scale Factor: In similar polygons, the ratio of one side of a polygon to the corresponding side of the other.
Example 4: What is the scale factor of △ABC to △XYZ? Write the similarity statement.
Solution: All the sides are in the same ratio. Pick the two largest (or smallest) sides to find the ratio.
For the similarity statement, line up the proportional sides. AB→XY,BC→XZ,AC→YZ, so △ABC∼△YXZ.
Example 5: ABCD∼AMNP. Find the scale factor and the length of BC.
Solution: Line up the corresponding sides. AB:AM, so the scale factor is 3045=23 or 32. Because BC is in the bigger rectangle, we will multiply 40 by 32 because it is greater than 1. BC=32(40)=60.
Example 6: Find the perimeters of ABCD and AMNP. Then find the ratio of the perimeters.
Solution: Perimeter of ABCD=60+45+60+45=210
Perimeter of AMNP=40+30+40+30=140
The ratio of the perimeters is 140:210, which reduces to 2:3.
Theorem 7-2: The ratio of the perimeters of two similar polygons is the same as the ratio of the sides.
In addition to the perimeter having the same ratio as the sides, all parts of a polygon are in the same ratio as the sides. This includes diagonals, medians, midsegments, altitudes, and others.
Example 7: △ABC∼△MNP. The perimeter of △ABC is 150, AB=32 and MN=48. Find the perimeter of △MNP.
Solution: From the similarity statement, AB and MN are corresponding sides. The scale factor is 3248=23. △ABC is the smaller triangle, so the perimeter of △MNP is 32(150)=225.
Know What? Revisited The baseball diamond is on the left and the softball diamond is on the right. All the angles and sides are congruent, so all squares are similar. All of the sides in the baseball diamond are 90 feet long and 60 feet long in the softball diamond. This means the scale factor is 9060=32.
- Questions 1-8 use the definition of similarity and different types of polygons.
- Questions 9-13 are similar to Examples 1, 5, 6, and 7.
- Questions 14 and 15 are similar to the Know What?
- Questions 16-20 are similar to Example 2.
- Questions 21-30 are similar to Examples 3 and 4.
For questions 1-8, determine if the following statements are true or false.
- All equilateral triangles are similar.
- All isosceles triangles are similar.
- All rectangles are similar.
- All rhombuses are similar.
- All squares are similar.
- All congruent polygons are similar.
- All similar polygons are congruent.
- All regular pentagons are similar.
△BIG∼△HAT. List the congruent angles and proportions for the sides.
- If BI=9 and HA=15, find the scale factor.
- If BG=21, find HT.
- If AT=45, find IG.
- Find the perimeter of △BIG and △HAT. What is the ratio of the perimeters?
- 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?
- 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.
- Find m∠E and m∠Q.
ABCDE∼QLMNP, find the scale factor.
- Find BC.
- Find CD.
- Find NP.
Determine if the following triangles and quadrilaterals are similar. If they are, write the similarity statement.
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