# How to Find Similarity and Ratios? (+FREE Worksheet!)

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

Solution:

Write the proportion and solve for the missing side.
$$\frac{Smaller \ triangle \ height}{Smaller \ triangle \ base} = \frac{Bigger \ triangle \ height}{Bigger \ triangle \ base}$$
$$⇒ \frac{180 \ cm}{90 \ cm }= \frac{x} {90 \ + \ 340 \ cm} ⇒=180 \ × \ 430= 90 × \ x ⇒ 77400= 90 \ x ⇒ x=\frac{77400}{90}=860$$ $$cm$$

### 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?

Solution:

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?

Solution:

Write the proportion and solve for the missing side.
$$\frac{Smaller \ triangle \ height}{Smaller \ triangle \ base}= \frac{Bigger \ triangle \ height}{Bigger \ triangle \ base}$$
$$⇒ \frac{160 \ cm}{90 \ cm }= \frac{x} {90+360 \ cm} ⇒160×450= 90 × x ⇒ 72000= 90 \ x ⇒ x=\frac{72000}{90}=800$$ $$cm$$

### 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?

Solution:

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

## Exercises for Finding Similarity and Ratios

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

1. $$\color{blue}{9}$$
2. $$\color{blue}{2}$$
3. $$\color{blue}{5}$$
4. $$\color{blue}{15}$$

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