Question

Find the length of $P Q$.

Answer 1

Determine the length of PQ:

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

To find \( PQ \), apply the law of cosines to triangle \( PAQ \):
\[ PQ^2 = AP^2 + AQ^2 - 2 \times AP \times AQ \times \cos(\angle PAQ) \]

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

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

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

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

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

Answer 2

Determine \( m \angle POQ \):

Because \( AP \) and \( AQ \) are tangents to the circle originating from point \( A \), the angle formed by these tangents at \( P \) and \( Q \) is equivalent to the angle at the circle's center subtended by chord \( PQ \). This relationship is expressed as:

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

Using the provided value for \( \angle PAQ \):

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

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

Answer 3

(a) Calculate the length of OA:
To determine the length of OA, we typically utilize the distance formula between two points in a coordinate system. Assuming O is the origin (0,0) and A has coordinates \((x_1, y_1)\), the length OA is given by:\[OA = \sqrt{x_1^2 + y_1^2}\]This formula computes the distance from the origin to point A, requiring only the coordinates of A.

(b) Determine the radius of the mirror:
The radius of a spherical mirror is directly related to its focal length. The process involves:1. Identify Given Information: Typically, either the focal length \( f \), or the object distance \( u \) and image distance \( v \) are provided.2. Apply the Mirror Formula: If \( u \) and \( v \) are known, use the mirror formula to find \( f \): \[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]3. Calculate the Radius: The radius of curvature \( R \) is twice the focal length: \[ R = 2f \]4. Example: Given \( u = 10 \, \text{cm} \) and \( v = 20 \, \text{cm} \): \[ \frac{1}{f} = \frac{1}{20} + \frac{1}{10} = \frac{3}{20} \] Therefore, \( f = \frac{20}{3} \, \text{cm} \). The radius is then: \[ R = 2 \times \frac{20}{3} = \frac{40}{3} \, \text{cm} \]Thus, the radius of the mirror is \( \boxed{\frac{40}{3} \, \text{cm}} \).

Summary:The length of OA is found using the distance formula. The radius of a mirror is calculated from its focal length, which can be determined using the mirror formula if object and image distances are known.