GANITA PRAKASH · GRADE 8 · PART II · CHAPTER 2
DOUBLE — Double a square using its diagonal
Think first. Use AI purposefully. Verify always.
DISCOVER
Start with a sketch and a prediction.
If you double a square’s side, do you double its area? Cut two identical squares along their diagonals and try a different arrangement.
Try on paper first. Use the model to test your idea before accepting the explanation.
Understand
Four congruent isosceles right triangles fit around their right-angle vertices to form a square. One original square contains two of these triangles. The new square therefore has twice the area, and each new side is an original diagonal.
Key Ideas
New area = 2 × original area
Doubling the side makes four times the area, not twice.
Visualise
Four triangles, one new square
Method
Cut each of two equal squares into two triangles. Place the four right-angle corners at one centre. Their hypotenuses form the boundary; each outer corner is 45° + 45° = 90°.
Worked Example
Given
A square with side 3 units.
To Find
Area of the square constructed on its diagonal
Property
Four congruent half-square triangles form the new square.
Solution
- The original area is = 9 square units.
- The new square contains twice as many congruent triangles.
Result
The new area is 18 square 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
A square has area 20. What is the area of the square on its diagonal?
Check & Feedback
Name the known quantities and target, establish the theorem conditions and check your result in the original relationship.
Textbook connection: Ganita Prakash · Grade 8 · Part II · Chapter 2 — The Baudhāyana–Pythagoras Theorem · printed page 34