GANITA MANJARI · GRADE 9 · PART I · CHAPTER 6

TRANSFORM — Squaring a Rectangle

Think first. Use AI purposefully. Verify always.

DISCOVER

Start with a question.

Can a rectangle be transformed into a square of equal area without simply estimating its side?

Understand

Baudhāyana’s construction uses a right triangle and the difference of two squares to construct a square of area ab.

Key Ideas

\frac{{(a + b)}^{2}}{4} - \frac{{(a - b)}^{2}}{4} = ab

The new square has the same area, not generally the same perimeter. Its side is $\sqrt{ab}$ , not $\frac{(a + b)}{2}$ .

Visualise

Start with rectangle ABCD: a = 8, b = 4.

Method

For a > b > 0, follow E, F, the auxiliary square AFGH, arc intersection K, perpendicular KP, and the square HPQS. Verify the length using the right triangle HKP.

Worked Example

Given

Rectangle sides 8 and 4 units.

To Find

Construct and verify the equal-area square.

Formula / Property / Theorem / Reasoning

${HP}^{2}$ = ${HK}^{2}$ − ${KP}^{2}$ .

Solution

HK = $\frac{(8 + 4)}{2}$ = 6.
KP = $\frac{(8 - 4)}{2}$ = 2.
${HP}^{2}$ = 36−4 = 32.

Result

Square side $\sqrt{32}$ = 4 $\sqrt{2}$ units; area 32 square units. Original numerical instance of Figure 6.30.

AIMS Method

A — Attempt independently. Make a prediction or start a sketch.
I — Investigate the idea. Identify the boundary, region and given conditions.
M — Make connections. Relate the figure to a known property or rearrangement.
S — Scrutinise your reasoning. Check conditions, units and whether the result is exact.

Think first. Use AI purposefully. Verify always.

Focused Practice

Use the construction lengths for a = 9 and b = 4. What is the new square’s side?

Your reasoning

Check & Feedback

Why is the result positive for a > b > 0?

Explain your conclusion

Textbook connection: Ganita Manjari · Grade 9 · Part I · Chapter 6 · 6.9 · Figure 6.30 and starred proof · printed pages 140–142.