Question

Find the length of $P Q$.

Answer 1

Find the length of PQ:

We are given that \( AP = AQ = 30 \, \text{cm} \), and the angle between the tangents, \( \angle PAQ = 60^\circ \).

To find \( PQ \), we can use the law of cosines in triangle \( PAQ \), where:
\[ PQ^2 = AP^2 + AQ^2 - 2 \times AP \times AQ \times \cos(\angle PAQ) \]

Substituting the given values:
\[ PQ^2 = 30^2 + 30^2 - 2 \times 30 \times 30 \times \cos(60^\circ) \]

Since \( \cos(60^\circ) = 0.5 \):

\(PQ^2 = 900 + 900 - 2 \times 30 \times 30 \times 0.5\)

\(PQ^2 = 900 + 900 - 900 = 900\)
\(PQ = \sqrt{900} = 30 \, \text{cm}\)

Thus, the length of \( PQ \) is \( 30 \, \text{cm} \).

Answer 2

Step 1: Understanding the given information:
We are asked to find \( m \angle POQ \), where \( AP \) and \( AQ \) are tangents to a circle from the point \( A \).
- Since \( AP \) and \( AQ \) are tangents, the angle between the tangents at points \( P \) and \( Q \) is equal to the angle at the center of the circle subtended by the chord \( PQ \).

Step 2: Relating the angle between tangents to the central angle:
The key property is that the angle between two tangents drawn from an external point to a circle is equal to twice the angle at the center of the circle subtended by the chord joining the points of contact. In other words:
\[ \angle POQ = 2 \times \angle PAQ \]

Step 3: Substituting the given value:
We are given that \( \angle PAQ = 60^\circ \). Substituting this into the equation above:
\[ \angle POQ = 2 \times 60^\circ = 120^\circ \]

Step 4: Conclusion:
Thus, the value of \( m \angle POQ \) is \( 120^\circ \).

Answer 3

Step 1: Understanding the given problem:
We are given two subparts to solve:
- (a) Find the length of \( OA \).
- (b) Find the radius of the mirror.
We need to apply relevant formulas based on the context of the problem. Since the specific values or details of the problem are not provided in your question, I will proceed with general guidelines.

Part (a) - Finding the length of \( OA \):
The length of \( OA \) typically refers to the distance from the center of the circle (O) to a point on the circle (A). If this is a geometry problem involving tangents and circles, and if \( A \) is the point of tangency, we can assume that \( OA \) is the radius of the circle.
- If \( OA \) is the radius, then \( OA = r \), where \( r \) is the radius of the circle.

Part (b) - Finding the radius of the mirror:
If the problem involves a mirror in the context of reflection or geometric optics, and the mirror is spherical, the radius can be determined by using relationships in optics or geometry.
For a spherical mirror, if the focal length \( f \) is known, the radius \( r \) is related to the focal length by the formula:
\[ r = 2f \] If additional information about focal length or other distances in the problem is provided, we can use these relations to find the radius of the mirror.

Conclusion:
- Part (a) depends on the given context, but the length of \( OA \) is likely the radius of the circle if \( A \) is the point of tangency.
- Part (b) assumes that the mirror is spherical and we use \( r = 2f \) if the focal length is provided.
Please provide more specific details to proceed with exact calculations if required.