Similar Triangle Find Missing Side- Worksheet and Solutions

Similar Triangles: Finding the Missing Side

Similar triangles are triangles that have the same shape but different sizes. Their corresponding angles are equal, and their corresponding sides are in the same proportion. That's it. No fluff.

This concept is useful in geometry, physics, architecture, and any field where you need to find distances you can't measure directly. If one triangle is a scaled-up version of another, you can use ratios to find unknown sides.

How to Identify Similar Triangles

You need to verify similarity before you can use it. Three main criteria:

• AA (Angle-Angle): Two angles of one triangle match two angles of another. If two angles are equal, the third must be too.
• SAS (Side-Angle-Side): Two sides are in proportion AND the included angle is equal.
• SSS (Side-Side-Side): All three sides of one triangle are in the same ratio as all three sides of another.

In most textbook problems, similarity is given or obvious from the diagram. Look for triangles that share an angle or sit inside each other.

The Ratio Method: Finding Missing Sides

Once you've confirmed triangles are similar, finding a missing side is straightforward. You set up a proportion using corresponding sides.

Step-by-Step Process

1. Identify the two similar triangles in your diagram.
2. List the known sides for each triangle.
3. Write the ratio: side from triangle 1 divided by its corresponding side from triangle 2.
4. Set this ratio equal to the ratio of the unknown side and its corresponding side.
5. Solve using cross multiplication.

Example 1

Triangle ABC is similar to Triangle DEF.

Given: AB = 6, BC = 8, AC = 10
DE = 3, DF = ?, EF = 5

Set up the proportion using corresponding sides:

AB/DE = BC/DF = AC/EF

Use AB/DE = AC/EF to verify:

6/3 = 10/5 → 2 = 2 ✓

Now find DF:

6/3 = 8/DF

Cross multiply: 6 × DF = 3 × 8

6 × DF = 24

DF = 4

Example 2

A 6-foot person casts a 4-foot shadow. A nearby tree casts a 20-foot shadow at the same time. How tall is the tree?

These form two similar right triangles (person + shadow, tree + shadow). The ratio of height to shadow length is the same for both.

6/4 = tree height/20

Cross multiply: 6 × 20 = 4 × height

120 = 4 × height

Height = 30 feet

Common Mistakes to Avoid

• Mixing up corresponding sides: Always match the correct sides between triangles. Don't pair a short side with a long side that doesn't correspond.
• Forgetting to reduce ratios: Simplify your ratios early. It makes calculations cleaner.
• Using the wrong triangle pair: Some diagrams have multiple triangles. Make sure you're comparing the right pair.
• Skipping the similarity check: Never assume triangles are similar. Prove it first.

Worksheet: Practice Problems

Try these problems. Answers are below.

Problem 1:
Triangle A: sides 5, 12, 13
Triangle B is similar to Triangle A with shortest side 15. Find the other two sides of Triangle B.

Problem 2:
A flagpole 8 meters tall casts a 6-meter shadow. A building casts a 45-meter shadow. How tall is the building?

Problem 3:
Two similar triangles have a side ratio of 3:7. If the smaller triangle has a side of 9 cm, what is the corresponding side in the larger triangle?

Problem 4:
In the diagram, Triangle PQR is similar to Triangle STU. PQ = 14, QR = 21, PR = 28. ST = 7, TU = 10.5. Find SU.

Solutions

Problem 1:
Scale factor = 15/5 = 3
12 × 3 = 36
13 × 3 = 39
Answer: 36, 39

Problem 2:
8/6 = x/45
8 × 45 = 6x
360 = 6x
x = 60 meters

Problem 3:
9 × (7/3) = 21 cm

Problem 4:
Scale factor = 14/7 = 2
SU = 10.5 × 2 = 21

Similar Triangles vs. Congruent Triangles

Don't confuse these. Congruent triangles are identical in size and shape. Similar triangles have the same shape but different sizes. Congruent is a special case of similar where the scale factor is exactly 1.

Getting Started: Quick Reference

When you encounter a similar triangle problem:

1. Read the problem. Identify what you know and what you need to find.
2. Confirm the triangles are similar. Look for the similarity statement (e.g., "Triangle ABC ~ Triangle DEF").
3. Set up your proportion. Match corresponding sides correctly.
4. Solve. Cross-multiply and divide.
5. Check your answer. Does it make sense given the scale factor?

That's the whole process. Practice with 10-15 problems and you'll have it locked down.