If a chord of length 24 cm is at a distance of 5 cm from centre, then find the radius of the circle.
13 cm
Finding the Radius of a Circle Given Chord Length and Distance
This problem involves understanding the relationship between a circle's radius, a chord's length, and the distance of the chord from the center of the circle. When a perpendicular line segment is drawn from the center of a circle to a chord, it bisects the chord. This creates a right-angled triangle where:
- The hypotenuse is the radius of the circle (\(r\)).
- One leg is the distance of the chord from the center (\(d\)).
- The other leg is half the length of the chord (\(\frac{L}{2}\)).
Applying the Pythagorean Theorem for Circle Geometry
We can use the Pythagorean theorem to relate these three quantities. 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. In our case, this translates to:
\(r^2 = d^2 + \left(\frac{L}{2}\right)^2\)
Step-by-Step Calculation of the Radius
We are given the following information:
- Length of the chord (\(L\)) = 24 cm
- Distance of the chord from the center (\(d\)) = 5 cm
First, let's find half the length of the chord:
\(\frac{L}{2} = \frac{24 \text{ cm}}{2} = 12 \text{ cm}\)
Now, substitute the values of \(d\) and \(\frac{L}{2}\) into the Pythagorean theorem formula:
\(r^2 = (5 \text{ cm})^2 + (12 \text{ cm})^2\)
Calculate the squares:
\(r^2 = 25 \text{ cm}^2 + 144 \text{ cm}^2\)
Add the values:
\(r^2 = 169 \text{ cm}^2\)
To find the radius \(r\), take the square root of both sides:
\(r = \sqrt{169 \text{ cm}^2}\)
\(r = 13 \text{ cm}\)
So, the radius of the circle is 13 cm.
Summary of Circle Dimensions
| Property | Value |
|---|---|
| Length of Chord (\(L\)) | 24 cm |
| Distance from Center (\(d\)) | 5 cm |
| Half Chord Length (\(L/2\)) | 12 cm |
| Radius (\(r\)) | 13 cm |
Revision Table: Key Concepts for Circle Problems
| Concept | Description | Formula/Relationship |
|---|---|---|
| Chord | A line segment connecting two points on the circle. | --- |
| Radius | Distance from the center to any point on the circle. | \(r\) |
| Distance from Center to Chord | The perpendicular distance from the center to the chord. Bisects the chord. | \(d\) |
| Pythagorean Theorem | Relates the sides of a right-angled triangle. | \(a^2 + b^2 = c^2\) (where \(c\) is the hypotenuse) |
Additional Information on Circle Properties
Understanding the properties of circles is fundamental in geometry. Here are a few more key facts:
- The longest chord in a circle is the diameter, which passes through the center. Its length is \(2r\).
- A perpendicular from the center to a chord always bisects the chord. This property is crucial for solving problems like the one above.
- Conversely, the line segment joining the center to the midpoint of a chord is perpendicular to the chord.
- Equal chords of a circle are equidistant from the center.
- Chords that are equidistant from the center are equal in length.
These properties help in solving various problems related to chords, radii, and distances in circles.
Important Questions from Circles, Chords and Tangents
If a tangent to a circle from a point P meets the circle at A with AP = 15 cm. Given that the radius of the circle is 8 cm, find the distance of P from the centre of the circle.
In a circle with a radius of 10 cm. XY and PQ are two parallel chords 12 cm and 16 cm in length, respectively. The two chords are situated on the opposite sides of the centre. The distance between the chords is:
Find the equation of the tangents to the circle x2 + y2 = 9 at x = 2.
Let \( C_1 \) and \( C_2 \) be two circles which do not externally touch and intersect each other and \( O_1 \), and \( O_2 \) be the centers of the circles, respectively. Let AB be the common transverse tangent to the circles such that P, Q are the points of tangency respectively to \( C_1 \), \( C_2 \). Let R be the point of intersection of \( O_1 O_2 \) and AB. If \( \angle PO_1R = 60^\circ \), find \( \angle QO_2R \) and \( \angle QRO_2 \) respectively.
Find the length of the direct common tangent to two circles with radii equal to $6$ cm and $18$ cm and the distance between their centers which is equal to $30$ cm.
