Question

Richa has a pencil, a compass, and a post card of size 12 cm x 7 cm. Which of the following diagrams can she draw in her notebook using only these objects?

• A. Circle of radius 13 cm
• B. Equilateral triangle of side 5 cm
• C. Square of side 6 cm
• D. Regular hexagon of side 10 cm

Hint:

For geometric construction problems, remember that a compass and straightedge can be used to create lengths that are not directly measurable. This often involves using geometric properties like the Pythagorean theorem to construct required lengths from known lengths.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This question tests our knowledge of geometric constructions using basic tools. The postcard serves as a straightedge and allows for measuring any length up to its dimensions (12 cm and 7 cm). The compass allows for drawing circles and arcs, and transferring lengths. We need to determine if each of the given figures can be constructed with these tools.

Step 2: Detailed Explanation:


A. Circle of radius 13 cm: To draw this, Richa needs to set her compass to a radius of 13 cm. She cannot measure 13 cm directly as the postcard's longest side is 12 cm. However, she can construct this length. Using the Pythagorean theorem, \(5^2 + 12^2 = 25 + 144 = 169 = 13^2\). She can measure 12 cm using the long side of the postcard and 5 cm using the short side (since 5<7). She can then construct a right-angled triangle with sides 5 cm and 12 cm. The hypotenuse of this triangle will be exactly 13 cm long. She can then set her compass to this length and draw the circle. Thus, (A) is possible.
B. Equilateral triangle of side 5 cm: Richa can measure a 5 cm length using the 7 cm side of the postcard. She can draw a line segment of 5 cm. Then, using the compass set to this length, she can draw arcs from both ends of the segment. The intersection of the arcs gives the third vertex of the equilateral triangle. Thus, (B) is possible.
C. Square of side 6 cm: She can measure a 6 cm length using the postcard. She can draw a 6 cm line segment. At one end, she can construct a perpendicular line (a standard compass-and-straightedge construction). She can then mark 6 cm on the perpendicular line and complete the square using the compass to transfer the 6 cm length. Thus, (C) is possible.
D. Regular hexagon of side 10 cm: She can measure 10 cm using the 12 cm side of the postcard. The side length of a regular hexagon is equal to the radius of its circumscribed circle. She can set her compass to 10 cm, draw a circle, and then, without changing the compass setting, mark six consecutive arcs along the circumference to find the vertices of the hexagon. Thus, (D) is possible.

Step 3: Final Answer:
All four geometric figures can be constructed using a pencil, compass, and the postcard as a ruler/straightedge.