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Applications of Distance Formula

Applications of the distance formula use side lengths found from coordinates to identify geometric properties such as equal sides, right triangles, rectangles, or squares.

In one line: Calculate first, classify later.

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Understand it

When vertices of a figure are given as coordinates, the distance formula helps find side lengths and sometimes diagonals. After finding lengths, we compare them with known properties: equal sides for isosceles triangle, three equal sides for equilateral triangle, Pythagoras check for right triangle, and equal diagonals with opposite sides equal for rectangles.

Example

If AB = 5, BC = 5, and AC = 8, then triangle ABC is isosceles because two sides are equal.

Formula and rule

Use d = √((x2 - x1)2 + (y2 - y1)2), then compare lengths with properties of triangles and quadrilaterals.

See it worked out

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

Worked example

  1. 1For A(0,0), B(4,0), C(4,3), lengths AB = 4, BC = 3, AC = 5.
  2. 2Since 42 + 32 = 52, triangle ABC is right-angled at B.

Method you can reuse

List vertices, find needed side lengths, compare lengths, apply the correct property, and write the conclusion with reason.

Visual tip

A rough labelled diagram of the vertices helps decide which pairs of points form sides and diagonals.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

If a triangle has side lengths 6, 8, and 10 from coordinate calculations, what can you conclude?

Think first, then show answer
The triangle is a right triangle because 62 + 82 = 36 + 64 = 100 = 102, so the largest side satisfies the Pythagoras condition.

Common wrong answer

Calling a triangle right-angled just because one side is 10 is wrong; the square relation with the largest side must be checked.

Don't mix this up

Students may calculate only one side and decide the figure type without comparing all required sides.

Exam tip

Write all relevant lengths clearly before naming the triangle or quadrilateral. Marks are often given for comparison, not only calculation.

Applications of Distance Formula diagram

Open full screen to read every label clearly.

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

Questions may ask students to prove that three given points form a right triangle, isosceles triangle, rectangle, square, or rhombus.
  1. 1

    Enough for 1 mark

    Applications of the distance formula use side lengths found from coordinates to identify geometric properties such as equal sides, right triangles, rectangles, or squares.

  2. 2

    Add for 2 marks

    Example: If AB = 5, BC = 5, and AC = 8, then triangle ABC is isosceles because two sides are equal.

  3. 3

    Add for 3 marks

    When vertices of a figure are given as coordinates, the distance formula helps find side lengths and sometimes diagonals.

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