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Verifying isosceles triangle using distance

A triangle is isosceles if two of its sides are equal in length.

In one line: Equal distances, not equal-looking lines.

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Understand it

On coordinate planes, side lengths should be checked with the distance formula rather than by eye. A sketch can mislead because the graph may not look perfectly balanced. If two calculated side lengths are equal, the triangle is isosceles.

Formula and rule

Use the distance formula for all three sides, then compare the results.

If any two sides are equal, the triangle is isosceles.

See it worked out

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

Worked example

For A(1, 2), B(5, 2), and C(3, 6), AB = 4.

  1. 1Also AC = √((3 - 1)2 + (6 - 2)2) = √20, and BC = √((3 - 5)2 + (6 - 2)2) = √20.
  2. 2Hence AC = BC, so the triangle is isosceles.

Method you can reuse

  1. 1Find all three side lengths.
  2. 2Simplify each distance carefully.
  3. 3Compare the three values.
  4. 4If two are equal, state isosceles.
  5. 5Mention the equal sides in the final answer.

Sketch it in your answer

Draw the triangle and label the side lengths after calculation. A quick sketch helps only after the exact distances are found.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

What should be true for a triangle to be isosceles when you use coordinates?

Think first, then show answer
Two side lengths must be equal after calculation, because the distance formula must confirm the equal sides.

Common wrong answer

A common wrong answer is saying the triangle is isosceles because it looks symmetric on the graph, even when the actual lengths are different.

Don't mix this up

Students often judge isosceles shape only from the picture and forget to calculate the side lengths.

Exam tip

Always compare the actual distances. Equal-looking sides on a sketch are not enough.

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

Show that a triangle with given coordinates is isosceles, or identify the equal sides from the distance calculation.
  1. 1

    Enough for 1 mark

    A triangle is isosceles if two of its sides are equal in length.

  2. 2

    Add for 2 marks

    Example: For A(1, 2), B(5, 2), and C(3, 6), AB = 4 and AC = BC = √20, so the triangle is isosceles.

  3. 3

    Add for 3 marks

    On coordinate planes, side lengths should be checked with the distance formula rather than by eye.

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