Circles Solved Examples (10 Mathematics)
Circle theorems involve angles, tangents, chords, and secants. These solved examples cover key circle properties including angle theorems, tangent-radius r
TL;DR: Circle theorems involve angles, tangents, chords, and secants. These solved examples cover key circle properties including angle theorems, tangent-rad…
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Circle theorems involve angles, tangents, chords, and secants. These solved examples cover key circle properties including angle theorems, tangent-radius r
Circles — Solved Numerical Examples (Step by Step)
Example 1: A chord of length 24 cm is at a distance of 5 cm from the center of a circle. Find the radius of the circle.
Solution: When a perpendicular is drawn from the center to a chord, it bisects the chord.
Let O be center, AB be the chord, and OM be perpendicular to AB.
Then AM = MB = 24/2 = 12 cm (half the chord)
OM = 5 cm (given distance)
In right triangle OMA:
OA² = OM² + AM² (by Pythagoras theorem)
OA² = 5² + 12²
OA² = 25 + 144
OA² = 169
OA = 13 cm (radius)
Example 2: Two tangents are drawn to a circle from an external point. The angle between them is 60°. Find the angle subtended by the two points of contact at the center.
Solution: Let P be the external point and A, B be the points of contact.
Let O be the center.
Tangent PA is perpendicular to OA, so angle OAP = 90°
Tangent PB is perpendicular to OB, so angle OBP = 90°
Given: angle APB = 60°
In quadrilateral OAPB:
angle OAP + angle APB + angle PBO + angle BOA = 360°
90° + 60° + 90° + angle BOA = 360°
240° + angle BOA = 360°
angle BOA = 120°
Example 3: The angle in a semicircle is always a right angle. If a semicircle has diameter AB = 10 cm and a point C is on the semicircle such that AC = 6 cm, find BC.
Solution: Since angle ACB is in a semicircle, angle ACB = 90°
AB is the diameter = 10 cm
AC = 6 cm
In right triangle ABC:
AB² = AC² + BC²
10² = 6² + BC²
100 = 36 + BC²
BC² = 64
BC = 8 cm
Example 4: A tangent and a radius to a circle at the point of contact make an angle of 90°. If a tangent at point P has length 8 cm and the distance from the external point to the center is 10 cm, find the radius.
Solution: Let O be center, P be point of contact, and Q be external point.
QP is tangent, so OP ⊥ QP and angle OPQ = 90°
Given:
QP = 8 cm (tangent length)
OQ = 10 cm (distance from external point to center)
In right triangle OPQ:
OQ² = OP² + QP²
10² = OP² + 8²
100 = OP² + 64
OP² = 36
OP = 6 cm (radius)
Example 5: If two circles touch each other externally, and their radii are 3 cm and 4 cm, find the distance between their centers.
Solution: When two circles touch each other externally, the distance between their centers equals the sum of their radii.
Let r1 = 3 cm and r2 = 4 cm
Distance between centers = r1 + r2 = 3 + 4 = 7 cm
Example 6: Two chords AB and CD of a circle intersect at point P inside the circle. If AP = 4 cm, PB = 6 cm, and CP = 3 cm, find PD.
Solution: By the intersecting chords theorem:
When two chords intersect inside a circle, the products of their segments are equal.
AP × PB = CP × PD
4 × 6 = 3 × PD
24 = 3 × PD
PD = 8 cm
Example 7: Find the area of the circle passing through the vertices of a square with side 4 cm.
Solution: The circle passing through all vertices of a square is the circumcircle of the square.
For a square with side a, the diagonal = a√2
The diagonal equals the diameter of the circumcircle.
Side of square = 4 cm
Diagonal = 4√2 cm
Diameter = 4√2 cm
Radius = 2√2 cm
Area = πr² = π(2√2)² = π × 8 = 8π cm²
Tips
- A perpendicular from the center to a chord always bisects the chord.
- A tangent to a circle is always perpendicular to the radius at the point of contact.
- Equal chords in a circle are equidistant from the center.
- Use the intersecting chords theorem and power of a point for problems involving intersecting lines.
Frequently Asked Questions
Is a line that touches the circle at only one point always a tangent?
Yes, a line that touches a circle at exactly one point is called a tangent. It is perpendicular to the radius at that point of contact.
Can two circles intersect at more than two points?
No, two distinct circles can intersect at most at two points. If they intersect at three or more points, they must be the same circle.
More Mathematics Solved Examples
- Real Numbers (HCF LCM Euclid)
- Pair of Linear Equations in Two Variables
- Statistics
- Linear Equations in Two Variables
- Heron's Formula
- Integrals (Indefinite and Definite)
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