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

Find \( m \angle POQ \):

Since \( AP \) and \( AQ \) are tangents to the circle from the point \( A \), the angle between the tangents at \( P \) and \( Q \) is equal to the angle at the center of the circle subtended by the chord \( PQ \). This means that:

\[ \angle POQ = 2 \times \angle PAQ \]

Substituting the given value of \( \angle PAQ \):

\[ \angle POQ = 2 \times 60^\circ = 120^\circ \]

Thus, \( m \angle POQ = 120^\circ \).

Answer 3

(a) Find the length of OA:
Let's begin by analyzing the given geometry or configuration that involves the points and lines related to the length of OA. Since the exact details are not provided, let's assume the following scenario to demonstrate the solution approach:

We are working with a right-angled triangle, where point \( O \) is the origin and \( A \) is a point on the coordinate plane. Suppose we are given that \( O \) and \( A \) are two points with known coordinates, and we need to find the length of the line segment \( OA \), which is the distance between these two points.

For example, assume that the coordinates of \( O \) are \( (0, 0) \) and the coordinates of \( A \) are \( (x_1, y_1) \). The length of the line segment \( OA \) can be calculated using the distance formula:
\[ OA = \sqrt{(x_1 - 0)^2 + (y_1 - 0)^2} \] Simplifying: \[ OA = \sqrt{x_1^2 + y_1^2} \] Thus, the length of \( OA \) is the square root of the sum of the squares of the coordinates of \( A \). This formula can be applied directly once the coordinates of point \( A \) are provided.

(b) Find the radius of the mirror:
Now, let's move on to finding the radius of the mirror. We assume that the mirror is a spherical or circular mirror, and we are given some information about the distances involved with the mirror. Let’s follow a step-by-step approach to solving this problem.

1. Given information:
- We are provided with the focal length \( f \) of the mirror (either directly or derived from the problem).
- We may also be given the object distance \( u \) and the image distance \( v \) formed by the mirror.

2. Mirror Formula:
The mirror formula connects the object distance \( u \), the image distance \( v \), and the focal length \( f \) of the mirror: \[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \] Here, \( f \) is the focal length of the mirror, \( v \) is the image distance, and \( u \) is the object distance. The signs of these quantities depend on the type of mirror (concave or convex) and the conventions used for distances.

3. Radius of the Mirror:
The focal length \( f \) is related to the radius of curvature \( R \) of the mirror by the formula: \[ f = \frac{R}{2} \] So, once we find the focal length \( f \), we can easily find the radius of the mirror using: \[ R = 2f \] If the focal length \( f \) is known, the radius of the mirror can be determined directly by doubling the focal length.

4. Example Calculation:
Suppose we are given the following values:
- Object distance \( u = 10 \, \text{cm} \)
- Image distance \( v = 20 \, \text{cm} \)

We can substitute these values into the mirror formula to find the focal length \( f \):
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} = \frac{1}{20} + \frac{1}{10} \] Simplifying: \[ \frac{1}{f} = \frac{1}{20} + \frac{2}{20} = \frac{3}{20} \] Thus: \[ f = \frac{20}{3} \, \text{cm} \] Now, we can find the radius \( R \) of the mirror: \[ R = 2f = 2 \times \frac{20}{3} = \frac{40}{3} \, \text{cm} \] Thus, the radius of the mirror is \( \boxed{\frac{40}{3} \, \text{cm}} \).

Conclusion:
In summary, we have used the distance formula to calculate the length of \( OA \), and applied the mirror formula to find the radius of the mirror. These methods can be adapted depending on the exact details provided in the problem.