A 15 cm long perpendicular is drawn from the centre of a circle to its 40 cm long chord. Find the radius of the circle.
25 cm
Finding the Radius of a Circle Using Chord and Perpendicular Distance
This problem involves a fundamental property of circles related to chords and the perpendicular drawn from the center. When a perpendicular is drawn from the center of a circle to a chord, it bisects the chord. This creates a right-angled triangle where:
- One leg is the perpendicular distance from the center to the chord.
- The other leg is half the length of the chord.
- The hypotenuse is the radius of the circle.
We are given the following information:
- Length of the perpendicular from the center to the chord = 15 cm
- Length of the chord = 40 cm
First, let's find the length of half the chord:
Now we have a right-angled triangle with legs of length 15 cm and 20 cm, and the hypotenuse is the radius (let's call it \(r\)). We can use the Pythagorean theorem to find the radius:
Let's calculate the squares:
Now substitute these values back into the Pythagorean theorem equation:
To find the radius \(r\), we need to take the square root of 625:
So, the radius of the circle is 25 cm.
This solution demonstrates how applying geometric properties and the Pythagorean theorem allows us to find unknown lengths in a circle.
Revision Table: Key Concepts in Circle Geometry
| Concept | Description | Relevance to Problem |
|---|---|---|
| Chord | A line segment connecting two points on a circle. | The problem provides the length of a chord. |
| Perpendicular from Center to Chord | A line segment from the center of the circle that intersects the chord at a 90° angle. | The problem provides the length of this perpendicular. It bisects the chord. |
| Radius | A line segment from the center of the circle to any point on the circle. | This is the value we need to find. It is the hypotenuse in the right triangle formed. |
| Pythagorean Theorem | In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides ($a^2 + b^2 = c^2$). | Used to relate the perpendicular, half-chord, and radius. |
Additional Information on Circle Properties
Understanding properties of circles is crucial for solving geometry problems. Here are a few more related concepts:
- Diameter: The longest chord passing through the center of the circle. It is twice the radius ($D = 2r$).
- Tangent: A line that touches the circle at exactly one point. The radius drawn to the point of tangency is perpendicular to the tangent.
- Arc: A portion of the circumference of a circle.
- Sector: A region of a circle enclosed by two radii and an arc.
- Segment: A region of a circle enclosed by a chord and an arc.
The property that a perpendicular from the center bisects a chord is a key theorem. Its converse is also true: the line segment joining the center to the midpoint of a chord is perpendicular to the chord. These properties are essential when dealing with lengths and distances within a circle.
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