# How to Find Similarity and Ratios

Two figures are similar if they have the same shape. Learn how to use ratios and proportions to solve similarity problems in few simple steps.

## Step by step guide to solve similarity and ratios problems

• Two or more figures are similar if the corresponding angles are equal, and the corresponding sides are in proportion.
• To solve the similarity problem, you usually need to create a proportion and solve for the unknown side.

### Similarity and Ratios – Example 1:

A girl $$180$$ cm tall, stands $$340$$ cm from a lamp post at night. Her shadow from the light is $$90$$ cm long. How high is the lamp post?

Solultion:

Write the proportion and solve for missing side.
$$\frac{Smaller \ triangle \ height}{Smaller \ triangle \ base} = \frac{Bigger \ triangle \ height}{Bigger \ triangle \ base}$$
$$⇒ \frac{90 \ cm}{180 \ cm }= \frac{90 \ + \ 340 \ cm}{x}⇒90 \ x=180 \ × \ 430⇒x= 860$$ cm

### Similarity and Ratios – Example 2:

A tree $$20$$ feet tall casts a shadow $$14$$ feet long. Jack is $$10$$ feet tall. How long is Jack’s shadow?

Solultion:

Write a proportion and solve for the missing number.
$$\frac{20}{14}=\frac{10}{x} → 20 \ x=10 \ × \ 14$$
$$20 \ x=140→x=\frac{140}{20}=7$$

### Similarity and Ratios – Example 3:

A tree$$160$$ cm tall, stands $$360$$ cm from a lamp post at night. Its shadow from the light is $$90$$ cm long. How high is the lamp post?

Solultion:

Write the proportion and solve for missing side.
$$\frac{Smaller \ triangle \ height}{Smaller \ triangle \ base}= \frac{Bigger \ triangle \ height}{Bigger \ triangle \ base}$$
$$⇒ \frac{90 \ cm}{160 \ cm }= \frac{90+360 \ cm}{x}⇒90x=160×450⇒x= 800$$ cm

### Similarity and Ratios – Example 4:

A tree $$32$$ feet tall casts a shadow $$12$$ feet long. Jack is $$6$$ feet tall. How long is Jack’s shadow?

Solultion:

Write a proportion and solve for the missing number.
$$\frac{32}{12}=\frac{6}{x} → 32x=6×12=72$$
$$32x=72→x=\frac{72}{32}=2.25$$

## Exercises for Finding Similarity and Ratios

Each pair of figures is similar. Find the missing side.

1. $$\color{blue}{12}$$
2. $$\color{blue}{2}$$
3. $$\color{blue}{5}$$
4. $$\color{blue}{15}$$

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