Exercise 8.4 Solutions

Construct: Breaking Rectangles (Page 199)

Q. Construct a rectangle that can be divided into 3 identical squares.

Solution:

1. Assume the side of the square is $s$.
2. The rectangle will have breadth $= s$ and length $= 3s$.
3. Example: Let side $= 2$ cm.
   • Breadth = 2 cm.
   • Length = 6 cm.
4. Construct a rectangle with sides 6 cm and 2 cm.
5. Mark points at 2 cm and 4 cm along the 6 cm side to divide it into three 2 cm squares.

Construct (Page 201)

Q1. A Square within a Rectangle Construct a rectangle of sides 8 cm and 4 cm. Construct a square inside such that they share the same center.

Solution:

1. Draw rectangle $ABCD$ ($8 \times 4$).
2. Draw diagonals $AC$ and $BD$ to find the center $O$.
3. To draw the inner square (e.g., side 2 cm):
   • From $O$, measure 1 cm up, 1 cm down, 1 cm left, 1 cm right? No.
   • Since it’s a square, its sides are parallel to the rectangle.
   • Half-side = 1 cm.
   • From $O$, go 1 cm vertically up/down to draw horizontal sides.
   • From $O$, go 1 cm horizontally left/right to draw vertical sides.
   • Draw lines through these points parallel to the outer sides.

Falling Squares / Shadings (Page 202) These are practical drawing exercises requiring precision.

• Key Strategy: Always start by drawing the main frame (square or rectangle) using the Ruler-Compass method (90° angles). Then use the ruler to measure distances along the sides to mark the smaller internal vertices.

Q4. Square with a Hole (Page 203) Hint: Think where the center of the circle should be. Solution: The center of the circle is the intersection of the diagonals of the square. Construct the square, draw faint diagonals to find the center, then use the compass to draw the circle.