Circle Geometry Problem
This problem involves finding the radius of a circle given the length of a tangent segment and the distance from the point of tangency to the circle's circumference.
Key Concepts
- Tangent Property: A radius drawn to the point of tangency is perpendicular to the tangent line.
- Pythagorean Theorem: In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (\(a^2 + b^2 = c^2\)).
Setup for Calculation
Let:
- 'r' be the radius of the circle (in cm).
- 'L' be the length of the tangent, given as 12 cm.
- 'd' be the distance from the external point to the circumference, given as 8 cm.
Consider a right-angled triangle formed by:
- The radius to the point of tangency (length = 'r').
- The tangent segment (length = 'L' = 12 cm).
- The line segment connecting the external point to the circle's center. The length of this segment is the distance from the point to the circumference plus the radius (length = d + r = 8 + r). This is the hypotenuse.
Radius Calculation
Using the Pythagorean theorem on the right-angled triangle:
\((\text{Hypotenuse})^2 = (\text{Tangent Length})^2 + (\text{Radius})^2\) \((d + r)^2 = L^2 + r^2\)Substitute the given values:
\((8 + r)^2 = 12^2 + r^2\)Expand the equation:
\(8^2 + 2(8)(r) + r^2 = 144 + r^2\) \(64 + 16r + r^2 = 144 + r^2\)Subtract \(r^2\) from both sides:
\(64 + 16r = 144\)Isolate the term with 'r':
\(16r = 144 - 64\) \(16r = 80\)Solve for 'r':
\(r = \frac{80}{16}\) \(r = 5\)Result
The radius of the circle is 5 cm.
Important Questions from Circles
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What is the area of minor segment ?
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If the centre of the circle passing through the origin is (3, 4), then the intercepts cut off by the circle on x-axis and y-axis respectively are
