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Geometric Construction of Irrationals

Geometric construction of irrationals is a method of locating values like √2, √3, and √5 on the number line using right triangles and the Pythagoras relation.

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Student-friendly explanation

A right triangle can create a hypotenuse of irrational length. By transferring that hypotenuse to the number line with a compass, the irrational number is marked accurately as a length.

How to write this in exams

  1. 1

    Start with the exact idea

    Geometric construction of irrationals is a method of locating values like √2, √3, and √5 on the number line using right triangles and the Pythagoras relation.

  2. 2

    Then show how to use it

    Draw the required base on the number line, construct a perpendicular of suitable length, join the starting point to the top point, calculate the hypotenuse using Pythagoras relation, and transfer that length with a compass.

  3. 3

    Add one concrete example

    To construct √2, take a unit segment on the number line, erect a perpendicular of length 1 at its endpoint, join the starting point to the top of the perpendicular, and transfer the hypotenuse to the number line.

  4. 4

    Avoid this incomplete answer

    For √5, using sides 1 and 5 is wrong because the hypotenuse would be √26, not √5.

Definition

Geometric construction of irrationals is a method of locating values like √2, √3, and √5 on the number line using right triangles and the Pythagoras relation.

Example

To construct √2, take a unit segment on the number line, erect a perpendicular of length 1 at its endpoint, join the starting point to the top of the perpendicular, and transfer the hypotenuse to the number line.

Rule to remember

Right-triangle rule: hypotenuse² = base² + perpendicular². Use it to obtain lengths like √2, √3, and √5.

Memory hook

Build the root as a hypotenuse, then swing it to the line.

Examples and method

Worked example

For √5, draw OA = 2 units on the number line, draw AB = 1 unit perpendicular to OA, join OB. Then OB² = 2² + 1² = 5, so OB = √5. Cut an arc with radius OB to mark √5.

Method to apply

Draw the required base on the number line, construct a perpendicular of suitable length, join the starting point to the top point, calculate the hypotenuse using Pythagoras relation, and transfer that length with a compass.

Diagram support

Draw a number line, a right triangle on a unit segment, and an arc from the starting point transferring the hypotenuse onto the number line.

How CBSE asks it

Students may be asked to construct √2, √3, or √5 on the number line and justify the construction.

Avoid common mistakes

Common confusion

Students often mark an approximate decimal directly instead of constructing the required length geometrically.

Common wrong answer

For √5, using sides 1 and 5 is wrong because the hypotenuse would be √26, not √5.

Exam tip

For construction questions, mention the right angle, the unit lengths used, and the hypotenuse obtained by Pythagoras relation.

Quick check

Why does a right triangle with perpendicular sides 1 and 1 give √2?

By Pythagoras relation, the hypotenuse squared is 1² + 1² = 2, so the hypotenuse length is √2.

Study the geometric construction of irrationals diagram carefully

Use the labelled diagram to keep geometric construction of irrationals clear in short answers and revision.

What this diagram makes clear

This diagram keeps the labels and direction of geometric construction of irrationals in the right order.

Where this helps in exams

Use this for labelled diagram work and short exam answers on geometric construction of irrationals.

Revision cue

Revise geometric construction of irrationals through the labels before writing the answer.

Answer writing and exam use

1-mark answer

Geometric construction of irrationals is a method of locating values like √2, √3, and √5 on the number line using right triangles and the Pythagoras relation.

2-mark answer

Geometric construction of irrationals is a method of locating values like √2, √3, and √5 on the number line using right triangles and the Pythagoras relation. Right-triangle rule: hypotenuse² = base² + perpendicular². Use it to obtain lengths like √2, √3, and √5. To construct √2, take a unit segment on the number line, erect a perpendicular of length 1 at its endpoint, join the starting point to the top of the perpendicular, and transfer the hypotenuse to the number line.

3-mark answer

A right triangle can create a hypotenuse of irrational length. By transferring that hypotenuse to the number line with a compass, the irrational number is marked accurately as a length. Right-triangle rule: hypotenuse² = base² + perpendicular². Use it to obtain lengths like √2, √3, and √5. For √5, draw OA = 2 units on the number line, draw AB = 1 unit perpendicular to OA, join OB. Then OB² = 2² + 1² = 5, so OB = √5. Cut an arc with radius OB to mark √5. Students may be asked to construct √2, √3, or √5 on the number line and justify the construction. For √5, using sides 1 and 5 is wrong because the hypotenuse would be √26, not √5.
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