Question

Three circles each of units radius intersects each other at \(P, Q_{1}\) and \(R . P, Q\) and \(R\) are the centres of the three circles. What is the sum of length of the arcs \(P Q, Q R\) and \(P R\) ? (a) \(\sqrt{3} \pi\) (b) \(2 \sqrt{2} \pi\) (c) \(\pi\) (d) none of these

Short answer

Answer: The sum of the arc lengths is π.

Step-by-step solution

Find the angle between P and Q centers

We can see that \(\triangle PQR\) is an equilateral triangle because each side has the same length (1+1=2). An equilateral triangle has all its angles equal to 60 degrees (or \(\frac{\pi}{3}\) radians). So, the angle between P and Q centers is \(\frac{\pi}{3}\).

Find the arc lengths

Arc length can be calculated using the formula: \(Arc\,length = r \cdot \theta\), where r is the radius of the circle and \(\theta\) is the angle in radians. Since the radii of the circles are 1 unit and the angles are the same (\(\frac{\pi}{3}\)), we have: \(Arc\,PQ = Arc\,QR = Arc\,PR = 1 \cdot \frac{\pi}{3} = \frac{\pi}{3}\).

Calculate the sum of the arc lengths

Now that we have the arc lengths, we can find the sum by adding them together: \(Sum = Arc\,PQ + Arc\,QR + Arc\,PR = \frac{\pi}{3} + \frac{\pi}{3} + \frac{\pi}{3} = \pi\). The sum of the arc lengths is \(\pi\). So the correct answer is (c) \(\pi\).

Key concepts

Circle

The concept of a circle is fundamental in geometry. A circle is defined as the set of all points in a plane that are at a fixed distance, known as the radius, from a central point called the center.

In different applications, the radius can vary which means circles can be different sizes, but they always maintain their property of symmetry.

• The distance across the circle through the center is called the diameter, which is twice the radius.
• The circumference is the total distance around the circle, given by the formula: \(C = 2\pi r\), where \(r\) is the radius.

Circles often appear in problems involving rotational symmetry or objects moving in a circular path. Understanding these properties is essential for solving various geometric problems, including those involving arc lengths and sectors of circles.

Arc length

An arc is a part of the circumference of a circle. The arc length is a measure of the distance along this curved line and can be calculated using a specific formula depending on the angle and radius of the circle

To find the arc length, you need to know:

• The radius of the circle \(r\).
• The angle \(\theta\) subtended by the arc at the center of the circle, measured in radians.

The formula for arc length is \(Arc\,length = r \cdot \theta\).

This formula comes in handy especially when determining parts of the circle less than a full rotation. Arcs can appear in various geometric constructions and real-world contexts, such as roads curved along a circular path or sectors in a pie chart.

Equilateral triangle

An equilateral triangle is a special type of triangle where all three sides are equal in length, as well as all three angles, which are each \(60^\circ\) or \(\frac{\pi}{3}\) radians.

Some key properties of equilateral triangles include:

• All sides and angles are congruent.
• Each angle measures \(60^\circ\), summing to \(180^\circ\) as is true for all triangles.
• It is also symmetric, with each of the sides having equal capacity to act as a base.

In geometry, equilateral triangles often serve as basic constructions or building blocks for more complex shapes, and understanding their properties is critical not just for problems involving arcs and circles, but for a wide range of geometric applications.

Radians

Radians are an alternative way to measure angles. Unlike degrees, which divide a circle into 360 parts, radians relate the angle to the radius of the circle.

The total angle around a circle is \(2\pi\) radians, which is equivalent to \(360^\circ\). Consequently:

• \(\frac{\pi}{2}\) radians is equivalent to \(90^\circ\).
• \(\pi\) radians is equivalent to \(180^\circ\).
• \(2\pi\) radians returns to the full circle, \(360^\circ\).

Radians are especially useful in higher mathematics, physics, and engineering, offering a natural and valuable relationship tied to the circumference of the circle itself.

Calculations using radians often have a cleaner relationship with trigonometric functions and thus are widely used in advanced applications.