Question

There are two circles intersecting each other. Another smaller circle with centre \(O\), is lying between the common region of two larger circles. Centres of the circle (ie., \(A, O\) and \(B\) ) are lying on a straight line. \(A B=16 \mathrm{~cm}\) and the radii of the larger circles are \(10 \mathrm{~cm}\) each. What is the area of the smaller circle? (a) \(4 \pi \mathrm{cm}^{2}\) (b) \(2 \pi \mathrm{cm}^{2}\) (c) \(\frac{4}{\pi} \mathrm{cm}^{2}\) (d) \(\frac{\pi}{4} \mathrm{~cm}^{2}\)

Short answer

Answer: The area of the smaller circle can be found using the formula: Area = \(π \cdot OC^2\), where OC is the radius of the smaller circle. To find the radius OC, we need to use the geometry of triangles and the facts about intersecting circles, as explained in the solution steps. After finding OC, we can plug it into the formula to compute the area of the smaller circle.

Step-by-step solution

Identifying the geometry of the problem

Let's label the points of intersection between the larger circles as P and Q. We can draw the line segments AC, BC, PA and BQ, where C is the center of the smaller circle. Now we have two isosceles triangles PAC and QBC because PA = AC and QB = BC because all are the radii of the two larger circles.

Finding angle AOB

First, we will find the angle AOB, which will help us to find the angles PAC and QBC. To do this, we can use the Cosine Rule on triangle AOB: cos(AOB) = \(\frac{AO^2 + BO^2 - AB^2}{2(AO)(BO)}\)

Finding the angles PAC and QBC

Since triangle PAC is an isosceles triangle, we know that angles PAC and APC are equal. To find angle PAC, we can use the rule that the sum of all angles in a triangle is equal to \(180^{\circ}\). Therefore, we have: PAC = \(\frac{180^{\circ} - AOB}{2}\). Similarly, QBC = \(\frac{180^{\circ} - AOB}{2}\).

Finding the radius of the smaller circle

Now, we will use the sine rule to find the radius OC of the smaller circle. In triangle PAC, we have: \(\frac{OC}{sin(PAC)} = \frac{AC}{sin(APOC)}\) Since PAC = QBC, we can also write the sine equation for triangle QBC: \(\frac{OC}{sin(QBC)} = \frac{BC}{sin(QBOC)}\) We can now combine these sine equations to get: OC = \(\frac{AC \cdot sin(APOC)}{sin(PAC)} = \frac{BC \cdot sin(QBOC)}{sin(QBC)}\)

Calculate the area of the smaller circle

Now that we have the length of OC (the radius), we can calculate the area of the smaller circle using the formula: Area = \(π \cdot OC^2\) By plugging in the value of OC from the previous step and computing the area, we can find the correct option among the given choices.

Key concepts

Intersecting Circles

When two circles intersect each other, they create a common region known as the lens or lune. This problem involves understanding how intersecting circles relate to each other geometrically. The points where the circles intersect are crucial in solving the given problem, as they can be used to construct line segments and triangles that will help in finding unknown lengths or areas.

In our exercise, the points of intersection are essential in forming isosceles triangles, which then become the foundation for applying further geometric rules and formulas. It is important to visualize these intersecting circles and the relationships between their centers and points of intersection to solve circle geometry problems effectively.

Isosceles Triangles

An isosceles triangle is defined by having at least two sides of equal length. In the context of intersecting circles, these equal sides are often radii of the circles. The importance of identifying isosceles triangles in circle geometry problems lies in the symmetrical properties they possess. These properties allow us to make deductions about angles and sides with more ease.

For instance, the apex angles can be found by subtracting the base angle from 180 degrees and dividing by 2. Recognizing isosceles triangles within a geometric figure gives students a powerful tool to break down complex diagrams into more manageable parts.

Cosine Rule

The Cosine Rule, also known as the Law of Cosines, is useful for calculating unknown lengths or angles in any triangle, not just right-angled triangles. This rule expresses the relationship between the lengths of the sides of a triangle and the cosine of one of its angles.

Application in Circle Geometry

In the solution of our problem, the Cosine Rule is used to find angle AOB, which is crucial in determining the area of the smaller circle. The formula is stated as:
\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]
where '\( a \)', '\( b \)', and '\( c \)' are sides of the triangle, and '\( C \)' is the angle opposite side '\( c \)'. In circle geometry, this helps us find the necessary angles when we have the lengths of the sides, which is often the case with the centers of circles forming a triangle.

Sine Rule

The Sine Rule, or the Law of Sines, is another trigonometric principle that relates the lengths of sides of a triangle to the sines of its angles. This rule is especially useful for triangles where we don't have a right angle.

Formula and Usage

The Sine Rule is given by:\[\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}\]
Where '\(a\)', '\(b\)', and '\(c\)' represent the triangle's sides, and '\(A\)', '\(B\)', and '\(C\)' the angles opposite those sides respectively. In our problem, the Sine Rule is used to equate the ratios of sides and sines of angles in the isosceles triangles formed by the intersecting circles, eventually leading to the determination of the smaller circle's radius.

Circle Area Calculation

Once the radius of a circle is known, calculating its area is a straightforward task. The area of a circle is given by the formula:\[\text{Area} = \pi r^2\]
where '\( r \)' is the radius of the circle. In the context of our exercise, after using geometric principles to determine the radius '\( OC \)' of the smaller circle, we apply this formula to arrive at the desired solution. This method of area calculation is fundamental to all circle geometry problems, and understanding it is essential for students to solve related exercises accurately.