1. Similar Shapes and Proportions
4.1
Similar Shapes and Proportions
How can you use ratios to determine if two figures are similar?

2. Texas Essential Knowledge and Skills
The student is expected to:
Proportionality—7.5.A
Generalize the critical attributes of similarity, including ratios within and between
similar shapes
Mathematical Processes
7.1.B
Use a problem-solving model that incorporates analyzing given information,
formulating a plan or strategy, determining a solution, justifying the solution, and
evaluating the problem-solving process and the reasonableness of the solution.

3. Warm Up
Find the cross products, and then tell whether the ratios are equal. 16 6 , 40 15 1.
240 = 240; equal 3 8 18 46 , 2.
138 = 144; not equal 8 9 , 24 27 3.
216 = 216; equal 28 12 , 42 18 4.
504 = 504; equal

4. Octahedral fluorite is a crystal found in nature
Octahedral fluorite is a crystal found in nature. It grows in the shape of an octahedron, which is a solid figure with eight triangular faces. The triangles in different-sized fluorite crystals are similar figures. Similar figures have the same shape but not necessarily the same size.

5. When naming similar figures, list the letters of the corresponding vertices in the same order. In the previous table ∆ABC ~ ∆DEF.
Writing Math

6. Matching sides of two or more polygons are called corresponding sides, and matching angles are called corresponding angles.
82◦
Corresponding angles D E F
Corresponding sides A B C
55◦
43◦

7. SIMILAR FIGURES Two figures are similar if
The measures of their corresponding angles are equal.
The ratios of the lengths of the corresponding sides are proportional.

8. A side of a figure can be named by its endpoints, with a bar above.
Without the bar, the letters indicate the length of the side.
Reading Math

9. Additional Example 1: Determining Whether Two Triangles Are Similar
Identify the corresponding sides in the pair of triangles. Then use ratios to determine whether the triangles are similar. E
AB corresponds to DE.
16 in
10 in A C
28 in
BC corresponds to EF.
4 in D
7 in
40 in F
AC corresponds to DF. B AB DE = ? BC EF = ? AC DF
Write ratios using the corresponding sides. 4 16 = ? 7 28 = ? 10 40
Substitute the length of the sides. 1 4 = ? 1 4 = ? 1 4
Simplify each ratio.
Since the ratios of the corresponding sides are equivalent, the triangles are similar.

10. Check It Out: Example 1
Identify the corresponding sides in the pair of triangles. Then use ratios to determine whether the triangles are similar. E
AB corresponds to DE.
9 in
9 in A C
21 in
BC corresponds to EF.
3 in D
7 in
27 in F
AC corresponds to DF. B AB DE = ? BC EF = ? AC DF
Write ratios using the corresponding sides. 3 9 = ? 7 21 = ? 9 27
Substitute the length of the sides. 1 3 = ? 1 3 = ? 1 3
Simplify each ratio.
Since the ratios of the corresponding sides are equivalent, the triangles are similar.

11. Additional Example 2: Determining Whether Two Four-Sided Figures are Similar
Tell whether the figures are similar.
The corresponding angles of the
figures have equal measure.
Write each set of
corresponding sides as a ratio.

12. Additional Example 2 ContinuedMN QR
MN corresponds to QR. NO RS
NO corresponds to RS. OP ST
OP corresponds to ST. MP QT
MP corresponds to QT.

13. Additional Example 2 Continued
Determine whether the ratios of the lengths of the corresponding sides are proportional. MN QR = ? NO RS OP ST MP QT
Write ratios using corresponding sides. 6 9 = ? 8 12 4 10 15
Substitute the length of the sides. 2 3 = ?
Simplify each ratio.
Since the ratios of the corresponding sides are equivalent, the figures are similar.

14. Tell whether the figures are similar.
Check It Out: Example 2
Tell whether the figures are similar.
100 m
80 m
60 m
47.5 m
80°
90°
125°
65° M P N O
400 m
320 m
190 m
240 m Q T R S
The corresponding angles of the
figures have equal measure.
Write each set of
corresponding sides as a ratio.

15. Check It Out: Example 2 Continued
80°
90°
125°
65° M P N O
400 m
320 m
190 m
240 m Q T R S MN QR
MN corresponds to QR. NO RS
NO corresponds to RS. OP ST
OP corresponds to ST. MP QT
MP corresponds to QT.

16. Check It Out: Example 2 Continued
Determine whether the ratios of the lengths of the corresponding sides are proportional.
100 m
80 m
60 m
47.5 m
80°
90°
125°
65° M P N O
400 m
320 m
190 m
240 m Q T R S MN QR = ? NO RS OP ST MP QT
Write ratios using corresponding
sides. 60
240 = ? 80
320
47.5
190
100
400
Substitute the length of the
sides. 1 4 = ?
Simplify each ratio.
Since the ratios of the corresponding sides are equivalent, the figures are similar.

17. ADDITIONAL EXAMPLE 1 Tell whether the triangles are similar.
Yes, the triangles are similar.

18. ADDITIONAL EXAMPLE 2
Molly made two sizes of tiles. Are the shapes of the tiles similar?
No, the shapes are not similar.

19. 4.1 LESSON QUIZ
7.5.A
Tell whether the shapes are similar. 1.
a rectangle with height 7 inches and length 12 inches, and a rectangle with height inches and length 15 inches
similar

20. Tell whether the shapes are similar.2.
not similar

21. Tell whether the shapes are similar.3.
not similar 4.
similar

22. 12 ft; since the triangles are similar, their
In the figure to the right, ΔABC is similar to ΔDAC.
What must the length of AC be? Explain.
12 ft; since the triangles are similar, their
corresponding sides are in proportion. so AC = 12.

23. How can you use ratios to determine if two figures are similar?
Sample answer:
If the measures of the corresponding angles are equal, check the ratios of the lengths of the corresponding sides. If the ratios are proportional, the shapes are similar.