C
CraftExam
high importancemedium8 min

Verifying isosceles triangle using distance

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

Practice This Concept

Learn the concept

Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

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

  2. 2

    Then show how to use it

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

  3. 3

    Add one concrete example

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

  4. 4

    Avoid this incomplete 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.

Definition

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

Example

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

Rule to remember

Use the distance formula for all three sides, then compare the results. If any two sides are equal, the triangle is isosceles.

Memory hook

Equal distances, not equal-looking lines.

Examples and method

Worked example

For A(1, 2), B(5, 2), and C(3, 6), AB = 4. Also AC = sqrt((3 - 1)^2 + (6 - 2)^2) = sqrt(20), and BC = sqrt((3 - 5)^2 + (6 - 2)^2) = sqrt(20). Hence AC = BC, so the triangle is isosceles.

Method to apply

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

Diagram support

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

How CBSE asks it

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

Avoid common mistakes

Common confusion

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

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.

Exam tip

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

Quick check

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

Two side lengths must be equal after calculation, because the distance formula must confirm the equal sides.

Answer writing and exam use

1-mark answer

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

2-mark answer

A triangle is isosceles if two of its sides are equal in length. Use the distance formula for all three sides, then compare the results. If any two sides are equal, the triangle is isosceles. For A(1, 2), B(5, 2), and C(3, 6), AB = 4 and AC = BC = sqrt(20), so the triangle is isosceles.

3-mark answer

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. Use the distance formula for all three sides, then compare the results. If any two sides are equal, the triangle is isosceles. For A(1, 2), B(5, 2), and C(3, 6), AB = 4. Also AC = sqrt((3 - 1)^2 + (6 - 2)^2) = sqrt(20), and BC = sqrt((3 - 5)^2 + (6 - 2)^2) = sqrt(20). Hence AC = BC, so the triangle is isosceles. Show that a triangle with given coordinates is isosceles, or identify the equal sides from the distance calculation. A common wrong answer is saying the triangle is isosceles because it looks symmetric on the graph, even when the actual lengths are different.
MCQ Quiz

Practice this concept with focused MCQs

Open the concept quiz intro first, review the test details, and then start a focused MCQ set from this concept only. Instant score and answer review are live now.

10 MCQs5 MinutesInstant Results
Practice This Concept

Help improve this page

Found something confusing, incorrect, or missing?

Practice This Concept