Figure It Out: Rectangles

Carpenter’s Problem (Page 83)

Question: A carpenter has one 8 cm strip. To make a rectangular frame, what should be the length of the other strip and where should they join?

Solution:

1. Length: The strips act as diagonals. In a rectangle, diagonals are of equal length.
   • Length of 2nd strip = 8 cm.
2. Intersection: Diagonals of a rectangle bisect each other.
   • They should be joined at the midpoint of both strips (4 cm from the ends).

Angles Between Diagonals (Page 85)

Question: If the angle between the diagonals is $60^\circ$ and the diagonals are equal and bisect each other, find the other angles.

Solution: Let the diagonals intersect at $O$. Given $\angle AOB = 60^\circ$. Since diagonals bisect and are equal, $\Delta AOB$ is an isosceles triangle ($OA = OB$).

1. Base angles of $\Delta AOB$: $\frac{180^\circ - 60^\circ}{2} = 60^\circ$.
2. So $\Delta AOB$ is equilateral.
3. Similarly, vertically opposite $\Delta DOC$ is equilateral.
4. Adjacent angle $\angle AOD = 180^\circ - 60^\circ = 120^\circ$.
5. Base angles of $\Delta AOD$: $\frac{180^\circ - 120^\circ}{2} = 30^\circ$.
6. Corner Angle $A = 60^\circ + 30^\circ = 90^\circ$.
   • This confirms the figure is a Rectangle.

Figure It Out (Page 90)

Question 1: Are there non-rectangles in the given figures?

• Figures (i), (ii), (iii) are standard rectangles.
• Figure (iv) is a Square ($4 \text{ cm} \times 4 \text{ cm}$).
• Answer: All are rectangles. A square is a special type of rectangle.