Question

Find the length of $P Q$.

Answer 1

Determine the length of PQ:

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

The law of cosines in triangle \( PAQ \) can determine \( PQ \):
\[ PQ^2 = AP^2 + AQ^2 - 2 \times AP \times AQ \times \cos(\angle PAQ) \]

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

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

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

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

The length of \( PQ \) is therefore \( 30 \, \text{cm} \).

Answer 2

Step 1: Understand the problem:
We need to find the measure of angle \( m \angle POQ \). We are given that \( AP \) and \( AQ \) are tangents to a circle originating from point \( A \).
- A property of tangents states that the angle formed by the tangents at the points of contact is related to the angle subtended at the center by the chord connecting these points.

Step 2: Apply the relevant geometric property:
The angle between two tangents drawn from an external point to a circle is twice the angle subtended at the center of the circle by the chord joining the points of tangency. This can be expressed as:
\[\angle POQ = 2 \times \angle PAQ\]

Step 3: Calculate the central angle:
Given that \( \angle PAQ = 60^\circ \). Substitute this value into the equation from Step 2:
\[\angle POQ = 2 \times 60^\circ = 120^\circ\]

Step 4: State the final answer:
Therefore, the measure of \( m \angle POQ \) is \( 120^\circ \).

Answer 3

Problem Decomposition:
The problem is divided into two parts:
- (a) Calculate the length of \( OA \).
- (b) Determine the radius of the mirror.
Appropriate formulas will be applied based on the problem's context. Specific numerical values are absent, so general approaches are presented.

Part (a) - Calculating \( OA \):
The length \( OA \) typically represents the distance from the center of a circle (O) to a point on its circumference (A). In geometrical contexts, particularly with tangents to circles, if A is the point of tangency, then \( OA \) is understood to be the radius of the circle.
- Therefore, \( OA = r \), where \( r \) denotes the circle's radius.

Part (b) - Determining the Mirror Radius:
For spherical mirrors in optical or geometrical problems, the radius can be found using established relationships. If the focal length \( f \) is known, the radius \( r \) is calculated using the formula:
\[r = 2f\]
The radius can be found if additional information, such as the focal length or other relevant distances, is provided.

Summary:
- Part (a): The length of \( OA \) is likely the radius of the circle, assuming A is a point of tangency and the context is geometric. Specific details are needed for confirmation.
- Part (b): Assuming a spherical mirror, the radius \( r \) is twice the focal length \( f \) (i.e., \( r = 2f \)), provided the focal length is given.
Specific problem details are required for precise calculations.