medium difficulty5 min

Using similarity to prove relations

Once two triangles are proved similar, you can use corresponding side ratios and equal angles to prove lengths, proportions, and parallel-line results.

In one line: Prove similarity once, then ratios do the rest.

Practice This Concept

Understand it

Similarity is not only a naming result. It becomes a tool for proving unknown lengths, ratios, and geometric relations inside a figure.

Example

If triangles ABC and DEF are similar, then AB/DE = BC/EF = AC/DF can be used to find a missing side or check a proportion.

Formula and rule

If triangles are similar, then corresponding sides are proportional and corresponding angles are equal.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

1If triangles ABC and DEF are similar with AB = 6 cm, DE = 9 cm and BC = 8 cm, then AB/DE = 2/3.
2So if EF is the matching side, BC/EF = 2/3 gives EF = 12 cm.

Method you can reuse

1Prove the triangles similar.
2Match the corresponding vertices.
3Write the ratio of matching sides.
4Substitute and solve the required relation.

Sketch it in your answer

Mark the two similar triangles clearly and keep the same vertex order while writing ratios.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Why is proving triangle similarity useful in geometry questions?

Think first, then show answer

Similarity is useful because it gives equal angles and proportional corresponding sides. That lets us prove missing lengths, side ratios, and other relations cleanly.

Common wrong answer

Mixing non-corresponding sides, such as AB with EF, which breaks the ratio comparison.

Don't mix this up

Students often stop after writing the similarity statement and do not use the corresponding side ratios.

Exam tip

After proving similarity, write only matching side ratios and keep the vertex order correct.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

A proof question may ask you to derive a side length, show two segments are proportional, or establish a line relation after proving similarity.

1
Enough for 1 mark
Once two triangles are proved similar, you can use corresponding side ratios and equal angles to prove lengths, proportions, and parallel-line results.
2
Add for 2 marks
Example: If triangles ABC and DEF are similar, then AB/DE = BC/EF = AC/DF can be used to find a missing side or check a proportion.
3
Add for 3 marks
Similarity is not only a naming result.

MCQ Quiz

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