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.**

### Download Similarity and Ratios Worksheet

## Answers

- \(\color{blue}{12}\)
- \(\color{blue}{2}\)
- \(\color{blue}{5}\)
- \(\color{blue}{15}\)