GANITA PRAKASH · GRADE 8 · PART II · CHAPTER 2
HALVE — Reverse the construction
Think first. Use AI purposefully. Verify always.
DISCOVER
Start with a sketch and a prediction.
Join consecutive side midpoints of a square. Predict the shape and the fraction of area enclosed before revealing the cut lines.
Try on paper first. Use the model to test your idea before accepting the explanation.
Understand
The four removed corner triangles are congruent right triangles with legs equal to half the original side. Their combined area is half the original square. The midpoint figure has four equal sides and right angles, so it is a square.
Key Ideas
Midpoint-square area = original area ÷ 2
Halving the side leaves one quarter of the area.
Visualise
Fold to the midpoints
Method
Mark all four midpoints and join consecutive ones. Each corner triangle is isosceles right. At each midpoint the inner angle is 180° − 45° − 45° = 90°; equal hypotenuses give equal sides.
Worked Example
Given
An outer square of side 8 units.
To Find
Area of its midpoint square
Property
Subtract the four congruent corner triangles.
Solution
- Each corner triangle has area = 8.
- Outer area = 64; removed area = 4 × 8 = 32.
Result
Midpoint-square area = 32 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
An outer square has area 50. What is its midpoint-square 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 36