Let's analyze and solve each question from the image one by one.

Question 1

In the first diagram, we have a circle with points $P$, $Q$, and $O$, where $O$ is the center, and the angle at $O$ is $120^\circ$. You are asked to find:

1. $\angle P$
2. $\angle Q$

Solution:

• In a circle, the angle subtended at the center by a chord (in this case, $POQ$) is twice the angle subtended on the circumference by the same chord. This is known as the angle at the center theorem.

  Given:

  • $\angle POQ = 120^\circ$

  By the theorem:

  • $\angle P = \angle Q = \frac{1}{2} \times \angle POQ = \frac{1}{2} \times 120^\circ = 60^\circ$

Thus, $\angle P = 60^\circ$ and $\angle Q = 60^\circ$.

Question 2

In this diagram, $PR$ is the diameter of the circle, and you are asked to:

1. Find $\angle PRQ$
2. Find $\angle PQS$
3. Find $\angle POS$

Solution:

• a. $\angle PRQ$: Since $PR$ is the diameter, by the angle in a semicircle theorem, the angle subtended by a diameter on the circle is always $90^\circ$.

  Therefore, $\angle PRQ = 90^\circ$.

• b. $\angle PQS$: From the diagram, $\angle PQS = 40^\circ$ (directly labeled).

• c. $\angle POS$: This angle can be calculated by using the sum of angles around point $O$. We know from the diagram that: $\angle POS = 180^\circ - 60^\circ = 120^\circ$ Therefore, $\angle POS = 120^\circ$.

Question 3

In the third diagram, the circle has radius $7$ cm and point $B$ lies outside the circle. The line segments $AB$ and $BC$ are tangents to the circle. You are asked to find:

1. The length of $AB$
2. The area of quadrilateral $OABE$
3. The length of $AC$

Solution:

• a. Length of $AB$: Given:

  • $OB = 25$ cm (distance from the center of the circle to point $B$).
  • The radius of the circle is $7$ cm.

  Using the Pythagorean theorem for the right triangle $OAB$, where $OB$ is the hypotenuse: $OB^2 = OA^2 + AB^2$ Substituting the values: $25^2 = 7^2 + AB^2 \implies 625 = 49 + AB^2 \implies AB^2 = 576 \implies AB = \sqrt{576} = 24 \, \text{cm}$

• b. Area of quadrilateral $OABE$: Quadrilateral $OABE$ consists of a right triangle $OAB$ and a rectangle $OABE$. The area of the triangle $OAB$ can be calculated as: $\text{Area of } OAB = \frac{1}{2} \times OA \times AB = \frac{1}{2} \times 7 \times 24 = 84 \, \text{cm}^2$

• c. Length of $AC$: Since $AB = AC$ (tangents drawn from an external point to a circle are equal), the length of $AC = 24$ cm.

Would you like more detailed steps or clarifications on any of the solutions? Here are some additional related questions:

1. What is the relationship between angles subtended by the same arc at different points?
2. How would you calculate the length of a tangent from a different point outside the circle?
3. Can you derive the general formula for the area of a sector of a circle?
4. How do we know that $\angle PRQ$ is $90^\circ$ from the diagram?
5. How can we use the Pythagorean theorem in problems involving tangents?

Tip: For problems involving circles, always remember the fundamental properties of tangents, chords, and angles subtended by arcs or diameters.