This problem requires finding the radius of a circle using the properties of tangents and the Pythagorean theorem.
Understanding the Geometry
Let 'O' be the center of the circle, 'Q' be the external point, and 'P' be the point where the tangent from Q touches the circle.
- The length of the tangent PQ = 21 cm.
- The distance from the center O to the point Q is OQ = 29 cm.
- The radius OP is perpendicular to the tangent PQ at the point of tangency P. This forms a right-angled triangle ΔOPQ.
- We need to find the radius, which is the length of OP.
Applying the Pythagorean Theorem
In the right-angled triangle ΔOPQ:
- The hypotenuse is OQ.
- The other two sides are OP (radius, 'r') and PQ (tangent length).
According to the Pythagorean theorem:
$ \OP^2 + PQ^2 = OQ^2 $
Calculation Steps
- Substitute the known values into the equation: $ r^2 + (21 \text{ cm})^2 = (29 \text{ cm})^2 $
- Calculate the squares: $ r^2 + 441 \text{ cm}^2 = 841 \text{ cm}^2 $
- Isolate $r^2$: $ r^2 = 841 \text{ cm}^2 - 441 \text{ cm}^2 $ $ r^2 = 400 \text{ cm}^2 $
- Find the radius 'r' by taking the square root: $ r = \sqrt{400 \text{ cm}^2} $ $ r = 20 \text{ cm} $
The radius of the circle is 20 cm.
Similar Questions
- PQ is a diameter of a circle whose centre is O. If a point R lies on the circle and $\angle RPO$ is $39^\circ$, then find the measure of $\angle RQP$.
- The points A, B and C lie on a circle with centre O. The $\angle \text{ACB} = 47.5^{\circ}$. What is the $\angle \text{AOB}$ on the minor $\widehat{\text{AB}}$?
- $1^{\circ}$ of latitude is equal to ___ km approximately.
- The radius of a circle is equal to the diagonal of a square. If the area of the square is 32 sq units, then the area of the circle is:
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- Two intersecting circles are said to be orthogonal if the angle between them is a/an _____.
A park is designed in the shape of a right-angled triangle, where the two shorter sides measure 13 m and 84 m. A circular track with a uniform width of 2 m is constructed such that its outer boundary coincides with the circumcircle of the triangular park. Determine the perimeter of the inner boundary of this circular track.
- If circumference and area of a circle are numerically equal then radius of the circle is:
- The angle of a sector is 30°. If its radius is 42 cm then the length of the arc of the sector is:
Important Questions from Circles
The sum of the radius and diameter of a circle is 84 cm. What is the circumference of this circle?
The maximum area of a right-angled triangle inscribed in a circle of radius r is
The tangent at a point C of a circle and diameter AB when extended intersect at D, if ∠DCA = 110°, then ∠CBA is equal to
The equation of a circle with diameters are 2x - 3y + 12 = 0 and x + 4y - 5 = 0 and area of 154 sq. units is
The internal center of similitude of two circles $ (x - 1)^2 + (y - 3)^2 = 4 $ and $ (x + 5)^2 + (y - 9)^2 = 16 $ is
