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medium difficulty5 min

Verifying right triangle using distance

A triangle is right-angled if its side lengths satisfy the Pythagorean relation with the longest side.

In one line: Longest side first, then Pythagoras.

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Understand it

In coordinate geometry, first calculate all three sides using the distance formula. Then identify the longest side and check whether its square equals the sum of the squares of the other two sides. This is the cleanest way to verify a right triangle from coordinates.

Example

For points (0, 0), (3, 0), and (3, 4), the side lengths are 3, 4, and 5, so 32 + 42 = 52.

Formula and rule

If the longest side is c and the other sides are a and b, then a2 + b2 = c2 for a right triangle.

See it worked out

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

Worked example

  1. 1For (0, 0), (3, 0), and (3, 4), the side lengths are 3, 4, and 5.
  2. 2Since 32 + 42 = 52, the triangle is right-angled.

Method you can reuse

  1. 1Find all three side lengths.
  2. 2Choose the longest side.
  3. 3Square all three lengths.
  4. 4Check whether the sum of the smaller squares equals the square of the longest side.
  5. 5State the conclusion clearly.

Sketch it in your answer

Plot the three points and mark the longest side mentally or on paper before comparing squares of lengths.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

What should be checked first when verifying a right triangle from coordinates?

Think first, then show answer
The longest side should be identified first, because only that side can act as the hypotenuse in the Pythagorean test.

Common wrong answer

A common wrong answer is picking the smallest side as the hypotenuse or checking the wrong pair of sides.

Don't mix this up

Students often choose the wrong side as the hypotenuse or forget to check the longest side first.

Exam tip

Always find the longest side before applying Pythagoras. The largest side must be tested as the hypotenuse candidate.

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

Verify whether a triangle with given coordinates is right-angled, or identify the hypotenuse from the calculated side lengths.
  1. 1

    Enough for 1 mark

    A triangle is right-angled if its side lengths satisfy the Pythagorean relation with the longest side.

  2. 2

    Add for 2 marks

    Example: For points (0, 0), (3, 0), and (3, 4), the side lengths are 3, 4, and 5, so 32 + 42 = 52.

  3. 3

    Add for 3 marks

    In coordinate geometry, first calculate all three sides using the distance formula.

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