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.

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

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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.

Download Similarity and Ratios Worksheet

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Answers

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

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