# 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.

### 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

### 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.
$$\frac{20}{14}=\frac{10}{x} → 20 \ x=10 \ × \ 14$$
$$20 \ x=140→x=\frac{140}{20}=7$$

### 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

### 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

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

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