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Verifying right triangle using distance

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

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Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

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

  2. 2

    Then show how to use it

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

  3. 3

    Add one concrete example

    For points (0, 0), (3, 0), and (3, 4), the side lengths are 3, 4, and 5, so 3^2 + 4^2 = 5^2.

  4. 4

    Avoid this incomplete answer

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

Definition

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

Example

For points (0, 0), (3, 0), and (3, 4), the side lengths are 3, 4, and 5, so 3^2 + 4^2 = 5^2.

Rule to remember

If the longest side is c and the other sides are a and b, then a^2 + b^2 = c^2 for a right triangle.

Memory hook

Longest side first, then Pythagoras.

Examples and method

Worked example

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

Method to apply

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

Diagram support

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

How CBSE asks it

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

Avoid common mistakes

Common confusion

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

Common wrong answer

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

Exam tip

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

Quick check

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

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

Answer writing and exam use

1-mark answer

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

2-mark answer

A triangle is right-angled if its side lengths satisfy the Pythagorean relation with the longest side. If the longest side is c and the other sides are a and b, then a^2 + b^2 = c^2 for a right triangle. For points (0, 0), (3, 0), and (3, 4), the side lengths are 3, 4, and 5, so 3^2 + 4^2 = 5^2.

3-mark answer

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. If the longest side is c and the other sides are a and b, then a^2 + b^2 = c^2 for a right triangle. For (0, 0), (3, 0), and (3, 4), the side lengths are 3, 4, and 5. Since 3^2 + 4^2 = 5^2, the triangle is right-angled. Verify whether a triangle with given coordinates is right-angled, or identify the hypotenuse from the calculated side lengths. A common wrong answer is picking the smallest side as the hypotenuse or checking the wrong pair of sides.
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