In a circle with radius 5 cm, a chord is at a distance of 3 cm from the centre. The length of the chord is:
8 cm
Finding the Length of a Chord in a Circle
This problem involves a circle, its radius, a chord, and the distance from the center to the chord. We need to find the length of the chord given the radius and the distance from the center.
Understanding the Geometry of the Circle and Chord
In any circle, a line segment drawn from the center perpendicular to a chord bisects the chord. This means it divides the chord into two equal parts. This perpendicular line segment, the radius drawn to an endpoint of the chord, and half of the chord form a right-angled triangle.
- The hypotenuse of this right triangle is the radius of the circle.
- One leg is the distance from the center to the chord.
- The other leg is half the length of the chord.
Applying the Pythagorean Theorem
We can use the Pythagorean theorem in the right-angled triangle formed. The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Let:
- \(r\) be the radius of the circle.
- \(d\) be the distance from the center to the chord.
- \(x\) be half the length of the chord.
According to the Pythagorean theorem:
\(d^2 + x^2 = r^2\)
Calculating Half the Chord Length
We are given:
- Radius, \(r = 5\) cm
- Distance from the center to the chord, \(d = 3\) cm
Substitute these values into the Pythagorean theorem equation:
\(3^2 + x^2 = 5^2\)
\(9 + x^2 = 25\)
Subtract 9 from both sides to isolate \(x^2\):
\(x^2 = 25 - 9\)
\(x^2 = 16\)
Take the square root of both sides to find \(x\):
\(x = \sqrt{16}\)
\(x = 4\) cm
So, half the length of the chord is 4 cm.
Calculating the Full Chord Length
Since \(x\) represents half the length of the chord, the full length of the chord is \(2 \times x\).
Chord Length = \(2 \times 4\) cm
Chord Length = \(8\) cm
Summary of Calculations
| Parameter | Value |
|---|---|
| Radius (r) | 5 cm |
| Distance from Center (d) | 3 cm |
| Half Chord Length (x) | 4 cm (calculated) |
| Full Chord Length | 8 cm (calculated) |
Therefore, the length of the chord is 8 cm.
Revision Table: Key Circle and Chord Concepts
| Concept | Description |
|---|---|
| Chord | A line segment connecting two points on the circumference of a circle. |
| Radius | A line segment from the center of the circle to any point on its circumference. |
| Distance from Center to Chord | The length of the perpendicular segment from the center to the chord. |
| Perpendicular Bisector Property | A perpendicular from the center to a chord bisects the chord. |
| Pythagorean Theorem | In a right triangle, \(a^2 + b^2 = c^2\). Used here with radius as hypotenuse and distance/half chord as legs. |
Additional Information: Related Circle Properties
Understanding chord properties is important in geometry. Here are a few related facts:
- The longest chord in a circle is the diameter, which passes through the center.
- Equal chords are equidistant from the center.
- Chords equidistant from the center are equal in length.
- If two chords intersect inside a circle, the products of the segments of each chord are equal (Intersecting Chords Theorem).
These properties are useful for solving various problems involving chords and circles.
Important Questions from Circles, Chords and Tangents
In a circle, a ten cm long chord is at a distance of 12 cm from the centre of the circle. The length of the diameter of the circle (in cm) is:
Chord AB of a circle of radius 10 cm is at a distance 8 cm from the centre O. If tangents drawn at A and B intersect at P., then the length of the tangent AP (in cm) is:
A chord 21 cm long is drawn in a circle of diameter 25 cm. The perpendicular distance of the chord from the centre is:
O is the centre of this circle. Tangent drawn from a point P, touches the circle at Q. If PQ = 24 cm and OQ = 10 cm, then what is the value of OP?
AB is the chord of a circle such that AB = 10 cm. If the diameter of the circle is 20 cm, then the angle subtended by the chord at the centre is ________.
