medium difficulty7 min

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.

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

Practice This Concept

Understand it

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.

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.

Formula and rule

Right-triangle rule: hypotenuse² = base² + perpendicular².

Use it to obtain lengths like √2, √3, and √5.

See it worked out

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

Worked example

For √5, draw OA = 2 units on the number line, draw AB = 1 unit perpendicular to OA, join OB.

1Then OB² = 2² + 1² = 5, so OB = √5.
2Cut an arc with radius OB to mark √5.

Method you can reuse

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.

Sketch it in your answer

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.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

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

Think first, then show answer

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

Common wrong answer

For √5, using sides 1 and 5.

Why it loses marks: The hypotenuse would be √26, not √5.

Don't mix this up

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

Exam tip

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