Circle Chord Length Calculation
This problem involves finding the length of a chord within a circle given its radius and its distance from the center. We can visualize a right-angled triangle formed by:
- The radius of the circle (hypotenuse).
- The distance from the center to the chord (one leg).
- Half the length of the chord (the other leg).
Geometric Setup
Let the radius of the circle be r = 15 cm.
Let the distance of the chord from the center be d = 9 cm.
Let half the length of the chord be x. The full length of the chord will be 2x.
Applying Pythagorean Theorem
According to the Pythagorean theorem in the right-angled triangle:
$r^2 = d^2 + x^2$
Substitute the given values:
$15^2 = 9^2 + x^2$
$225 = 81 + x^2$
Calculating Half Chord Length
Solve for $x^2$:
$x^2 = 225 - 81$
$x^2 = 144$
Find the value of x:
$x = \sqrt{144}$
$x = 12 \text{ cm}$
Determining Full Chord Length
The length of the chord is twice the value of x:
Chord Length = $2x$
Chord Length = $2 \times 12$ cm
Chord Length = 24 cm
Final Answer
The length of the chord is 24 cm.
Similar Questions
- A circle that has a radius of 5 centimeters, and there is a chord measuring 8 centimeters in length. What is the distance from the center of the circle to the chord?
- 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 $x$
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.
If a chord of length 24 cm is at a distance of 5 cm from centre, then find the radius of the circle.
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.
