Similar Figures

Key Concepts

• Understand similarity
• Complete a similarity transformation
• Identify similar figures

Understand Similar Figures  

Similar figures have the same shape but different sizes.  

In more mathematical language, two figures are similar if their corresponding angles are harmonious, and the ratios of the lengths of their equivalent sides are equal. 

Some similar figures are given below 

6.7.1 Understanding similarity  

Similarity: 

two objects are similar if they have the same shape, or one has the same shape as the mirror image of the other.  

More precisely, one can be obtained from the other by uniformly scaling (enlarging or reducing), possibly with additional translation, rotation and reflection. 

Example: 

Charmin graphed the below figures. How can she tell whether the figures are similar? 

Sol: 

Step 1:  

The orientation of figure Y is reversed from that of figure X, so a reflection over the y-axis is needed. 

Step 2:  

Figure Y has sides that are half as long as figure X, so a dilation with a scale factor of 1/2 is needed. 

Step 3: 

Figure B is moved to the right and above figure A, so a translation is needed. 

Step 4:  

Figure Y is moved above figure X, so a translation is needed. 

Step 5:  

A sequence of transformations that can accomplish this is a dilation by a scale factor of 1/2, centred at the origin, followed by the reflection. 

(x, y) —> (-x, y) 

followed by the translation 

(x, y) —> (x, y + 5) 

They are similar.  

6.7.2 Complete a similarity transformation. 

Two examples of similarity transformations are  

1. a translation and reflection and 

2. a reflection and dilation. 

Example: 

Describe a similarity transformation that maps trapezoid PQRS to trapezoid WXYZ. 

Solution  

—  

QR falls from left to right, and  

—  

XY   

rises from left to right. If you reflect  

trapezoid PQRS in the y-axis as  

shown, then the image, trapezoid  

P′Q′R′S′, will have the same  

orientation as trapezoid WXYZ 

Trapezoid WXYZ appears to be about one-third as large as trapezoid P′Q′R′S′.  

Dilate trapezoid P′Q′R′S′ using a scale factor of
1313

.

The vertices of trapezoid P, Q, R, and S match the vertices of trapezoid WXYZ.  

So, a similarity transformation that maps trapezoid PQRS to trapezoid WXYZ involves a reflection in the y-axis followed by dilation with a scale factor of 1.

1313

. 

6.7.3. Identify similar figures 

Two figures are considered ” similar” if they have the same shape, corresponding congruent angles (meaning the angles in the same places of each shape are the same), and equal scale factors.  

Equal scale factors mean that the lengths of their corresponding sides have a matching ratio. 

Example: 

Is ABC similar to DEF? Explain. 

Sol: 

AB−AB-

    corresponds to

 DE  −  DE  – 

BC− BC- 

corresponds

EF−EF-

 [Equation]  corresponds to [Equation] 

Write ratios using the corresponding sides. 

        ?          ? 

[Equation]  = [Equation]  =[Equation]   

Substitute the length of the sides. 

         ?? 

  [Equation] = [Equation] =[Equation] 

Simplify each ratio. 

[Equation] = [Equation] = [Equation] 

Since the ratio of the corresponding sides is equal, the triangles are similar. 

Exercise:

1. Is AMNO similar to APQO? Explain

2. How are similar figures related by a sequence of transformations?
3. The two triangles are similar what is the scale of the dilation from the small triangle to the large one?

4) Identify the similar transformation that could map delta ABC to DEF.

5) What do similar figures have in common?

6) Find the sequence of similarity transformation that maps the first figure to the second figure. Write the coordinate notation for each transformation.

7) What makes two similar figures?

8) From the graph below, determine if the two figures are similar. Then write the sequence of transformations that map the figure1 to figure 2.

9) When a figure is translated, reflected, rotated, or dilated in the plane, is the image always similar to the original figure? Explain your reasoning.

10) Give two examples of similarity transformations.

What Have We learned:

• Understand similar figures
• Understand similarity
• Complete a similarity transformation
• Identify similar figures