Question

In the adjoining figure \(O\) is the centre of the circle. The radius OP bisects a rectangle \(A B C D\), at right angle. \(D M=N C=2 \mathrm{~cm}\) and \(A R=S B=1 \mathrm{~cm}\) and \(K S=4 \mathrm{~cm}\) and \(O P=5 \mathrm{~cm}\). What is the area of the rectangle? (a) \(8 \mathrm{~cm}^{2}\) (b) \(10 \mathrm{~cm}^{2}\) (c) \(12 \mathrm{~cm}^{2}\) (d) None of these

Short answer

a) 8 cm² b) 10 cm² c) 12 cm² d) 14 cm²

Step-by-step solution

Find the length of AR and DM

Given that AR = SB = 1 cm and DM = NC = 2 cm, we already have half the width and length of the rectangle.

Find the length of SK in terms of OP

Since OP bisects the rectangle and is a radius of the circle, it is perpendicular to KD and CB. Thus, by Pythagorean theorem, we can calculate the length of SK, using the values of KS = 4 cm and OP = 5 cm: \[SK = \sqrt{OP^2 - KS^2} = \sqrt{5^2 - 4^2} = \sqrt{9} = 3 \mathrm{~cm}\]

Calculate the width and length of the rectangle

Since we now have the lengths of AR and SK, we can calculate the width and length of the rectangle. The length, AD, is equal to the sum of DM and SK: AD = DM + SK = 2 cm + 3 cm = 5 cm. The width, AB, is equal to the sum of AR and BS: AB = AR + SB = 1 cm + 1 cm = 2 cm.

Calculate the area of the rectangle

The area of a rectangle is calculated by multiplying the length by the width. In this case, the area of rectangle ABCD is: Area = AD × AB = 5 cm × 2 cm = 10 cm². The answer is (b) 10 cm².

Key concepts

Circle Properties

A circle is a simple shape in geometry where every point on its boundary is equidistant from a fixed point called the center. This fixed point is crucial for defining many properties of the circle.
Some key properties of a circle are:

• Radius: The distance from the center to any point on the circle. In this exercise, the radius OP is important as it bisects the rectangle.
• Diameter: The longest distance across the circle, passing through the center. It's twice the radius.
• Circumference: The total distance around the circle, calculated as \(2\pi r\).
• Area: The space inside the circle, calculated as \(\pi r^2\).

Understanding these properties helps clarify how the circle interacts with other shapes like rectangles in geometric problems.

Rectangle Area Calculation

The area of a rectangle is a basic calculation in geometry. It helps us understand the size of the rectangle.
The formula for finding the area is simple: multiply the rectangle's length by its width.
In this exercise, the length AD and the width AB were found to be 5 cm and 2 cm, respectively. Thus, the area of the rectangle is:
\[ \text{Area} = \text{Length} \times \text{Width} = 5\, \text{cm} \times 2\, \text{cm} = 10\, \text{cm}^2\]
Finding these measurements relies on understanding and using the fixed dimensions provided in the problem. It highlights how knowing precise lengths allows us to determine area efficiently.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry dealing with right-angled triangles. It's expressed as:
\[ a^2 + b^2 = c^2\]
where \(c\) is the hypotenuse, and \(a\) and \(b\) are the other two sides.
In the context of this exercise, the radius OP and segment KS were parts of a right triangle. Here, OP is the hypotenuse with \(OP = 5\, \text{cm}\) and \(KS = 4\, \text{cm}\). By rearranging the theorem equation, we found SK as:
\[ SK = \sqrt{OP^2 - KS^2} = \sqrt{5^2 - 4^2} = 3\, \text{cm}\]
Utilizing this theorem is essential for calculating lengths in geometric problems involving right triangles.

Bisecting Radius

A bisecting radius divides an object into two equal parts. In geometric problems, it often involves symmetry and right angles.
In this exercise, radius OP bisects the rectangle at a right angle, meaning it splits the rectangle into two equal segments. The intersected line segments are of equal length, and this property helps us verify measurements like SK.

• Perpendicular Line: When a bisecting radius creates a right angle with another line, it ensures symmetrical division.
• Symmetry: Makes the analysis and calculation more straightforward as each side mirrors the other.

Understanding bisecting lines aids in solving geometrical puzzles and ensuring accurate measurements.