In the given circle, diameter AB is extended to meet chord CD extended at P. If the lengths of the line segments AP, CP and DP are 12 cm, 8 cm and 6 cm, respectively, what is the length of the radius of the circle?
To solve this problem, we will use the Power of a Point Theorem, which states that for a point P outside a circle, the power of the point is equal to the product of the distances from P to two points on the circle.
In this case, the theorem implies that the product of the whole segment AP and its external segment PB (since AB is a diameter) is equal to the product of CP and DP.
Given:
- \(AP = 12 \, \text{cm}\)
- \(CP = 8 \, \text{cm}\)
- \(DP = 6 \, \text{cm}\)
Using the Power of a Point Theorem, we have: \(AP \times PB = CP \times DP\)
Substituting the known values: \(12 \times PB = 8 \times 6\)
Solving for PB: \(12 \times PB = 48\) \(PB = \frac{48}{12} = 4 \, \text{cm}\)
Since AB is the diameter of the circle, and ∠APB is a right angle (by Thales' theorem), the radius of the circle is half of AB: \(AB = AP - PB = 12 - 4 = 8 \, \text{cm}\)
Therefore, the radius of the circle is: \(\frac{AB}{2} = \frac{8}{2} = 4 \, \text{cm}\)
Thus, the correct answer is 4 cm.
Similar Questions
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Important Questions from Circles, Chords and Tangents
- A circular coin of radius 1 cm is allowed to roll freely on the periphery over a circular disc of radius 10 cm. If the disc has no movement and the coin completes one revolution rolling on the periphery over the disc and without slipping, then what is the number of times the coin rotated about its centre ?
There are two concentric circles. The radii of the two circles are 100 m and 110 m respectively. A wheel of radius 30 cm rolls on the smaller circle and another wheel rolls on the larger circle. After they have completed one revolution, it is found that the two wheels rolled equal number of times
on their respective axes. What is the radius of the other wheel ?- The length of a tangent drawn to a circle of radius 4 cm from a point 5 cm away from the center of the circle is:
- If SA and SB are tangents to a circle from an outside point S, such that SA = 12 cm and angle ASB = 60°. What will the length of chord AB be?
- In a circle, two chords of lengths x and y subtend the angles of $60^{\circ}$ and $90^{\circ}$ at the center respectively. Which of the following is correct in this scenario?
