Exercise 8.5 Solutions

Explore (Page 204)

Q. How should the rectangle be constructed so that the diagonal divides the opposite angles into equal parts?

Answer: If the diagonal divides the right angle ($90^\circ$) into two equal parts, each part is $45^\circ$.

• If the angle between the diagonal and the side is $45^\circ$, the triangle formed is an Isosceles Right-Angled Triangle.
• This implies the adjacent sides are equal.
• Therefore, the rectangle must be a Square.

Construct (Page 205)

Q1. Construct a rectangle where the diagonal divides opposite angles into $60^\circ$ and $30^\circ$.

Solution:

1. Draw a line $AB$ (arbitrary length).
2. At $A$, construct a $90^\circ$ angle (perpendicular).
3. We want the diagonal to make $60^\circ$ or $30^\circ$ with $AB$. Let’s assume $\angle CAB = 60^\circ$.
4. Construct a $60^\circ$ angle at $A$.
5. Extend this line until it hits the perpendicular from $B$ (if constructed that way) or set a specific length.
   • Correction based on text method: Draw triangle $ABC$ with $\angle B = 90^\circ$ and $\angle A = 60^\circ$.
   • Complete the rectangle by drawing parallel lines from $C$ and $A$.

Q2. Construct a rectangle where one side is 5 cm and diagonal is 7 cm.

Solution: See the method in “Topics: Diagonals”.

1. Draw base 5 cm.
2. Draw perpendicular at one end.
3. Cut arc of 7 cm from the other end.