Consider the following for the next two (02) items that follow: A chord of length l of a circle makes an angle 90° at the centre of the circle.
What is the area of the minor segment ?
Understanding the Problem: Finding the Area of a Minor Segment
The question asks for the area of the minor segment of a circle. We are given the length of a chord, denoted by \(l\), and the fact that this chord creates a 90° angle at the center of the circle. To find the area of the minor segment, we need to calculate the area of the sector formed by the central angle and subtract the area of the triangle formed by the chord and the two radii.
Given Information
- Length of the chord = \(l\)
- Central angle subtended by the chord, \(\theta = 90^\circ\)
Relationship Between Chord Length and Radius
When a chord subtends a 90° angle at the center of a circle, the triangle formed by the two radii connecting the endpoints of the chord to the center and the chord itself is a right-angled triangle. Let the radius of the circle be \(r\). The two radii forming the 90° angle are the legs of the right triangle, and the chord is the hypotenuse.
Using the Pythagorean theorem in this right-angled triangle:
\(r^2 + r^2 = l^2\)
\(2r^2 = l^2\)
\(r^2 = \frac{l^2}{2}\)
This gives us the relationship between the radius squared (\(r^2\)) and the chord length squared (\(l^2\)). We will use \(r^2\) in the area calculations.
Calculating the Area of the Sector
The area of a sector with central angle \(\theta\) (in degrees) and radius \(r\) is given by the formula:
\(A_{sector} = \frac{\theta}{360^\circ} \times \pi r^2\)
In this case, \(\theta = 90^\circ\), so the area of the sector is:
\(A_{sector} = \frac{90^\circ}{360^\circ} \times \pi r^2 = \frac{1}{4} \pi r^2\)
Now, substitute the value of \(r^2\) in terms of \(l\):
\(A_{sector} = \frac{1}{4} \pi \left(\frac{l^2}{2}\right) = \frac{\pi l^2}{8}\)
Calculating the Area of the Triangle
The triangle formed by the chord and the two radii is a right-angled triangle with legs of length \(r\). The area of a right triangle is given by \(\frac{1}{2} \times base \times height\).
In this case, base = \(r\) and height = \(r\), so the area of the triangle is:
\(A_{triangle} = \frac{1}{2} \times r \times r = \frac{1}{2} r^2\)
Now, substitute the value of \(r^2\) in terms of \(l\):
\(A_{triangle} = \frac{1}{2} \left(\frac{l^2}{2}\right) = \frac{l^2}{4}\)
Calculating the Area of the Minor Segment
The area of the minor segment is the difference between the area of the sector and the area of the triangle:
\(A_{segment} = A_{sector} - A_{triangle}\)
\(A_{segment} = \frac{\pi l^2}{8} - \frac{l^2}{4}\)
To match the format of the options, we can factor out a common term. Let's factor out \(\frac{l^2}{4}\):
\(A_{segment} = \frac{l^2}{4} \left(\frac{\pi l^2 / 8}{l^2 / 4} - \frac{l^2 / 4}{l^2 / 4}\right)\)
\(A_{segment} = \frac{l^2}{4} \left(\frac{\pi l^2}{8} \times \frac{4}{l^2} - 1\right)\)
\(A_{segment} = \frac{l^2}{4} \left(\frac{4\pi}{8} - 1\right)\)
\(A_{segment} = \frac{l^2}{4} \left(\frac{\pi}{2} - 1\right)\)
This result matches one of the given options.
Checking the Options
Let's compare our derived area of the minor segment, \(\frac{l^2}{4} \left(\frac{\pi}{2} - 1\right)\), with the given options:
- Option 1: \(\frac{l^2}{2}\left(\pi-\frac{1}{2}\right)\)
- Option 2: \(\frac{l^2}{4}\left(\pi-\frac{1}{2}\right)\)
- Option 3: \(\frac{l^2}{4}\left(\frac{\pi}{2}-1\right)\)
- Option 4: \(\frac{l^2}{2}\left(\frac{\pi}{2}-\frac{1}{2}\right)\)
Our calculated area matches Option 3.
| Concept | Formula Used |
|---|---|
| Pythagorean Theorem | \(a^2 + b^2 = c^2\) |
| Area of Sector | \(\frac{\theta}{360^\circ} \times \pi r^2\) |
| Area of Right Triangle | \(\frac{1}{2} \times base \times height\) |
| Area of Minor Segment | Area of Sector - Area of Triangle |
Revision Table: Key Concepts in Circle Geometry
| Term | Definition | Related Formula (if any) |
|---|---|---|
| Chord | A straight line segment connecting two points on the circumference of a circle. | |
| Radius (r) | A line segment from the center of the circle to any point on its circumference. | |
| Central Angle (\(\theta\)) | The angle formed by two radii at the center of the circle. | |
| Sector | A region of a circle bounded by two radii and the intercepted arc. | \(A = \frac{\theta}{360^\circ} \times \pi r^2\) (for angle in degrees) |
| Segment | A region of a circle bounded by a chord and the arc subtended by the chord. Minor segment is the smaller area, major segment is the larger area. | \(A_{segment} = A_{sector} - A_{triangle}\) (for minor segment) |
| Area of Triangle | The space enclosed by three straight sides. | \(A = \frac{1}{2} \times base \times height\) or \(A = \frac{1}{2}ab\sin(C)\) |
Additional Information: Special Cases and Relationships
Understanding the relationship between the chord length, radius, and central angle is crucial in circle geometry problems involving segments and sectors.
- For a central angle of 90°, the triangle formed is a right-angled isosceles triangle. The chord length \(l\) is related to the radius \(r\) by \(l = r\sqrt{2}\), or \(r = \frac{l}{\sqrt{2}}\). Squaring the radius gives \(r^2 = \frac{l^2}{2}\), which was used in our calculations.
- For a central angle of 180°, the chord is a diameter (\(l=2r\)). The segment is a semicircle. The triangle area is 0 (a degenerate triangle). The sector area is \(\frac{180}{360}\pi r^2 = \frac{1}{2}\pi r^2\). Area of segment = \(\frac{1}{2}\pi r^2\).
- For a central angle of 60°, the triangle formed is an equilateral triangle with side length equal to the radius, so \(l=r\). The area of the triangle is \(\frac{\sqrt{3}}{4}r^2 = \frac{\sqrt{3}}{4}l^2\). The area of the sector is \(\frac{60}{360}\pi r^2 = \frac{1}{6}\pi r^2\). Area of segment = \(\frac{1}{6}\pi r^2 - \frac{\sqrt{3}}{4}r^2\).
These special cases highlight how the geometry of the inscribed triangle changes with the central angle, affecting the segment area calculation.
Similar Questions
An equilateral triangle ABC and a scalene triangle DBC are inscribed in a circle on same side of the arc. what is ∠BDC equal to?
What is the length of QC ?
What is the relation between x and y ?
If y = 15, then what is ∠ACB equal to ?
If ∠ABD = θ, then what is sin θ equal to ?
What is the radius of the circle ?
What is the radius of the circle ?
- What is the area of the major segment ?
What is the length of the chord a unit circle which subtends an angle 2θ at the centre, where θ < 45°?
Consider the following statements :
1. If two chords AB and AC of a circle are equal, then the centre of the circle lies on the angle bisector of angle CAB.
2. If two concentric circles are intersected by a line at A, B, C and D respectively, then AC = BD.
Which of the above statements is/are correct?
Important Questions from Circles, Chords and Tangents
AB is a chord of a circle with centre O and P is any point on the circle. If ∠APB = 112°, then what is the measure of ∠OAB ?
The distance between the centres of two circles is 24 cm. If the radius of the two circles are 4 cm and 8 cm, then what is the sum of the lengths (in cm) of the direct common tangent and the transverse common tangent?
An equilateral triangle ABC and a scalene triangle DBC are inscribed in a circle on same side of the arc. what is ∠BDC equal to?
What can be the maximum number of common tangent which can be drawn to two non-intersecting circles?
There are two identical circles of radius 10 cm each. If the length of the direct common tangent is 26 cm, then what is the length (in cm) of the transverse common tangent?
