Understanding the Geometry
We are given a circle with a radius of 5 cm and a chord of length 8 cm. We need to find the distance from the circle's center to this chord.
Key Geometric Principle
The line segment drawn from the center of a circle perpendicular to a chord bisects the chord. This creates a right-angled triangle.
- The hypotenuse of this triangle is the circle's radius.
- One leg is half the length of the chord.
- The other leg is the distance from the center to the chord, which we need to find.
Applying the Pythagorean Theorem
Let:
- $r$ = radius = 5 cm
- $c$ = chord length = 8 cm
- $d$ = distance from the center to the chord
The half-length of the chord is $\frac{c}{2} = \frac{8 \text{ cm}}{2} = 4 \text{ cm}$.
According to the Pythagorean theorem, in the right-angled triangle formed:
$r^2 = (\frac{c}{2})^2 + d^2$
Calculation
- Substitute the known values into the equation:
$5^2 = 4^2 + d^2$
- Calculate the squares:
$25 = 16 + d^2$
- Isolate $d^2$:
$d^2 = 25 - 16$
$d^2 = 9$
- Solve for $d$ by taking the square root:
$d = \sqrt{9}$
$d = 3 \text{ cm}$
The distance from the center of the circle to the chord is 3 cm.
Similar Questions
- A circle of radius of 12 cm, two radii are drawn such that the length of the chord connecting their endpoints is 12 cm. What is the area of the minor segment formed?
- The line segment from the center of a circle to the midpoint of a chord is 10 cm long. If the radius is 26 cm, what is the length of the chord?
- A chord of a circle subtends an angle of 60° at the center. What is the angle subtended at a point on the circle in the same segment?
- In a circle, there is a chord measuring $20\text{ cm}$ that is located $15\text{ cm}$ away from the center. What is the approximate radius of the circle?
- A tangent is drawn to a circle with a radius of $5\text{ cm}$. At the point of tangency, what is the angle between the tangent and the radius?
- The radii of three concentric circles are in arithmetic progression. If the innermost radius is 7 cm and the outermost is 21 cm, what is the radius of the middle circle?
- If a chord subtends an angle of $75^{\circ}$ at the center, the angle it subtends at the circumference on the same side is:
- A chord subtends an angle of 70° at the center of a circle. What is the measure of the angle subtended by the same chord at any point on the remaining part of the circle?
- Chord AB and chord CD are equal and subtend angles of $50^\circ$ and $x^\circ$ at the center respectively. Find $x$
- A chord is drawn in a circle with a radius of 15 cm. If the distance of the chord from the center is 9 cm, what is the length of the chord?
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:
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:
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?
