Question

The length of a tangent from a point A at a distance 7.5 cm from the centre of the circle is 6 cm. Find the radius of the circle.

Hint:

Recognizing Pythagorean triplets can save time. This problem involves a scaled version of the 3-4-5 triplet. If you divide the given lengths by 1.5, you get 4 (for 6 cm) and 5 (for 7.5 cm). The missing side must be 3, and scaling it back by multiplying by 1.5 gives \(3 \times 1.5 = 4.5\) cm.

Answer 1

Step 1: Understanding the Concept:
A tangent to a circle is a line that touches the circle at exactly one point. The radius drawn to this point of tangency is always perpendicular to the tangent. This forms a right-angled triangle, which allows us to use the Pythagorean theorem.
Step 2: Key Formula or Approach:
Let O be the center of the circle, A be the external point, and T be the point of tangency. The radius OT is perpendicular to the tangent AT. The triangle \(\triangle OTA\) is a right-angled triangle with the hypotenuse being OA.
By Pythagorean theorem: \( OA^2 = OT^2 + AT^2 \).
Step 3: Detailed Explanation:
Let \(r\) be the radius of the circle.
Given:
Distance of point A from the center, OA = 7.5 cm.
Length of the tangent from point A, AT = 6 cm.
Radius of the circle, OT = \(r\).
In the right-angled triangle \(\triangle OTA\):
\[ OA^2 = OT^2 + AT^2 \] \[ (7.5)^2 = r^2 + (6)^2 \] \[ 56.25 = r^2 + 36 \] \[ r^2 = 56.25 - 36 \] \[ r^2 = 20.25 \] \[ r = \sqrt{20.25} \] \[ r = 4.5 \] Step 4: Final Answer:
The radius of the circle is 4.5 cm.