GANITA PRAKASH · GRADE 8 · PART II · CHAPTER 2
COMBINE — Combine two different squares
Think first. Use AI purposefully. Verify always.
DISCOVER
Start with a sketch and a prediction.
Join squares of sides a and b, with b ≥ a. Can two cuts and a rearrangement turn their combined area into one square?
Try on paper first. Use the model to test your idea before accepting the explanation.
Understand
The source construction keeps the five-sided piece V and relocates two congruent right triangles W and X to T and U. The resulting outer boundary has four equal hypotenuses and four right angles. No area is added or lost.
Key Ideas
Area(T + U + V) = Area(W + X + V)
Equal total area alone does not prove the new boundary is a square; check its sides and angles.
Visualise
Move the pieces; preserve the area
Two cuts separate W and X from the five-sided piece V. Total area: + = 45.
All four new sides are congruent hypotenuses. The two acute angles of a right triangle are complementary, so each new corner is 90°. This establishes a square before using its area formula.
Method
Mark a width-a strip at the right of the larger square. Draw its diagonal, then the second cut from the top-left corner of the smaller square to the strip’s bottom-left corner. Move W and X without stretching. Check the complementary acute angles at each new corner.
Worked Example
Given
Squares with sides 3 and 4 units.
To Find
Area of the square made by the source dissection
Property
Rearrangement preserves area; the boundary is a square by congruence and complementary angles.
Solution
- The two original areas are 9 and 16.
- The central piece stays fixed; the two right triangles are rigidly moved.
Result
The combined square has area 25 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
What must be checked in addition to preserving total area?
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 44