Let's solve each problem step by step:

Question 8: Circle Geometry

In the figure, point $M$ is the center of the circle. The angles $x$, $y$, and a given angle of $52^\circ$ are labeled. The goal is to find $x$ and $y$.

Part (a): Find angle $x$

• Since $M$ is the center of the circle, the two segments from $M$ to the circumference are radii. This makes the triangle formed by these two radii an isosceles triangle.
• The sum of the internal angles of a triangle is always $180^\circ$. Let's denote the angle at the center of the circle (where the radii meet) as $\theta$.
• The angle opposite the $52^\circ$ angle at the base of the isosceles triangle is also $52^\circ$.
• Therefore, $\theta + 52^\circ + 52^\circ = 180^\circ$, so: $\theta = 180^\circ - 104^\circ = 76^\circ.$ Thus, the central angle $\theta = 76^\circ$, and $x = 76^\circ$.

Part (b): Find angle $y$

• $y$ is part of the cyclic quadrilateral formed inside the circle.
• The sum of opposite angles in a cyclic quadrilateral is always $180^\circ$.
• Given that one of the opposite angles is $52^\circ$, we have: $y + 52^\circ = 180^\circ.$ Thus, $y = 180^\circ - 52^\circ = 128^\circ$.

Question 9: Right Triangle Check

We are given a triangle with sides 9 cm, 7 cm, and 6 cm. To check if the triangle is a right triangle, we can apply the Pythagorean theorem: $a^2 + b^2 = c^2,$ where $a$ and $b$ are the shorter sides, and $c$ is the hypotenuse (longest side).

Here, $a = 6$ cm, $b = 7$ cm, and $c = 9$ cm. Let's check if this satisfies the Pythagorean theorem: $6^2 + 7^2 = 9^2,$ $36 + 49 = 81,$ $85 \neq 81.$ Since the sides do not satisfy the Pythagorean theorem, this triangle is not a right triangle.

Summary of Solutions:

• $x = 76^\circ$
• $y = 128^\circ$
• The triangle is not a right triangle.

Do you need further clarification or details on any of the steps?

Follow-up questions:

1. What is the significance of opposite angles in a cyclic quadrilateral?
2. How would you approach finding the area of the triangle in Question 9?
3. Can you apply the Pythagorean theorem to any triangle? Why or why not?
4. What is the relationship between the angles in an isosceles triangle?
5. How does the central angle theorem help in solving circle-related problems?

Tip:

Always remember that the sum of the angles in any triangle is $180^\circ$. This is a crucial fact for solving geometric problems involving triangles.