GANITA PRAKASH · GRADE 8 · PART II · CHAPTER 2

ROOT — Discover √2

Think first. Use AI purposefully. Verify always.

DISCOVER

Start with a sketch and a prediction.

A unit square has area 1. What do you know about the area of the square on its diagonal, even before you know that diagonal’s length?

Try on paper first. Use the model to test your idea before accepting the explanation.

Understand

The diagonal is a positive length whose square is 2. We call that length $\sqrt{2}$ . More generally, an isosceles right triangle with equal legs a has hypotenuse a $\sqrt{2}$ .

Key Ideas

$c^{2}$ = $2 a^{2}$

The formula needs an isosceles right triangle, not just any isosceles triangle.

Visualise

The diagonal as a length

Original side: 1

Method

Use the geometric doubling result, then the relation between a square’s area and its positive side length. Keep the radical exact.

Worked Example

Given

An isosceles right triangle with equal legs 12.

To Find

Length of its hypotenuse

Property

$c^{2}$ = $2 a^{2}$ from square doubling.

Solution

$c^{2}$ = 2 × $12^{2}$ = 288.
c = $\sqrt{288}$ = 12 $\sqrt{2}$ .
$16^{2}$ < 288 < $17^{2}$ , so 16 < c < 17.

Result

The hypotenuse is exactly 12 $\sqrt{2}$ units.

AIMS Method

A — Attempt independently
Sketch and predict before using a formula.
I — Investigate the idea
Separate given facts from visual assumptions.
M — Make connections
Connect area, length, algebra and geometric constructions.
S — Scrutinise your reasoning
Check each hypothesis and justify the conclusion.

Think first. Use AI purposefully. Verify always.

Focused Practice

An isosceles right triangle has hypotenuse $\sqrt{72}$ . Find each leg.

Your reasoning

Continue with Roots and bounds →

Check & Feedback

Name the known quantities and target, establish the theorem conditions and check your result in the original relationship.

Explain your conclusion

Textbook connection: Ganita Prakash · Grade 8 · Part II · Chapter 2 — The Baudhāyana–Pythagoras Theorem · printed page 37