GANITA PRAKASH · GRADE 8 · PART II · CHAPTER 2

APPLY — Model, construct and justify

Think first. Use AI purposefully. Verify always.

DISCOVER

Start with a sketch and a prediction.

A lotus stem rises 1 unit above the water. Its tip touches the water 3 units away when tilted. What stays the same, and where is the right angle?

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

Understand

Translate the situation into a right triangle before using the theorem. For the lotus model, a straight stem of unchanged length pivots at a fixed root; its initial position is perpendicular to level water. Rhombi, square diagonals and grid squares also reveal useful right triangles.

Key Ideas

$x^{2}$ + $3^{2}$ = (x + 1) $^{2}$

Use half-diagonals in a rhombus. Distinguish a square’s area from its side length.

Visualise

Model the lotus, then solve

Horizontal reach (height above water = 1): 3

Assumptions: level water; initially perpendicular, straight stem; fixed root; unchanged stem length.

Build a square on grid points

Horizontal step p: 1

Vertical step q: 1

Area = $1^{2}$ + $1^{2}$ = 2 square units. Perpendicular side steps: $(1, 1)$ and (-1,1).

Area 3 is impossible: both integer step magnitudes would be at most 1, whose squared sum is at most 2.

Method

Draw and label the model, identify known quantities and the target, state assumptions and solve. Preserve radicals and check the result in the original equation.

Worked Example

Given

A rhombus has diagonals 24 and 70 units.

To Find

Length of each rhombus side

Property

Rhombus diagonals bisect each other at right angles.

Solution

Half-diagonals are 12 and 35.
A side is the hypotenuse: $s^{2}$ = $12^{2}$ + $35^{2}$ = 1369.
s = 37.

Result

Each side is 37 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

Find the area of an equilateral triangle of side 6.

Your reasoning

Continue with Applications →

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 52