From a point A, a tangent line is drawn to a circle of radius 7 units. From the same point, a secant is drawn to the circle which cuts the circle at points B and C, with B closer to A than C. Given the length of the tangent from A to the circle is 21 units and the length of segment AB is 14 units, find the length of BC.

Question

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Given:
- Radius of circle (r) = 7 units
- Length of tangent from A to circle = 21 units
- Length of AB = 14 units

We need to find BC.

Step 1: Use the tangent-secant theorem (Power of a Point theorem):
The square of the length of the tangent from point A to the circle equals the product of the lengths of the secant segments from A, i.e.,

(Tangent length)^2 = AB × AC

Step 2: Substitute the known values:
21^2 = 14 × AC
441 = 14 × AC
AC = 441 / 14 = 31.5 units

Step 3: Since AC = AB + BC, we find BC:
BC = AC - AB = 31.5 - 14 = 17.5 units

Step 4: Check the options:
17.5 units is option 4.

However, the answer given in the options is 16.5 as correct. Let's re-check calculations.

Re-check:
21^2 = 441
AB = 14
AC = 441 / 14 = 31.5
BC = 31.5 - 14 = 17.5

So the correct length of BC is 17.5 units.

Therefore, the correct answer is option 4: 17.5 units.

Answer

Answer

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