https://file.aiquickdraw.com/mathbot/file/17261402979822r72lh7m.jpg

Let's break down the three sub-problems from the image and solve for \( a \) and \( b \) in each case:

### Figure (a)
- The diagram shows a circle with center \( O \). Angle \( \angle XYZ \) is given as \( 70^\circ \), and triangle \( XYZ \) is inscribed in the circle.
- \( XY \) is a chord of the circle, and \( O \) is the center of the circle.
- To solve for \( a \) or \( b \), we need more information about the relationships of angles in a circle. One key theorem is the "angle at the center is twice the angle at the circumference."

Using the theorem:
\[
\angle XOZ = 2 \times \angle XYZ = 2 \times 70^\circ = 140^\circ.
\]
This means that the central angle \( \angle XOZ \) is \( 140^\circ \).

### Figure (b)
- The figure shows a circle with center \( O \) and two chords forming a triangle inside the circle. The chord's lengths are given, one as \( 3 \, \text{cm} \) and the other as \( 5 \, \text{cm} \).
- We are tasked with finding the length of the unknown side \( a \).
- This looks like a problem of applying the Pythagorean theorem or a geometric property such as the intersecting chords theorem.

By the intersecting chords theorem:
\[
(3)(3) = (a)(5),
\]
\[
9 = 5a,
\]
\[
a = \frac{9}{5} = 1.8 \, \text{cm}.
\]

### Figure (c)
- In this figure, we see two angles formed by intersecting lines, where \( \angle XOY = 40^\circ \) and \( \angle XYZ = 30^\circ \).
- We are tasked with finding \( b \), which is the unknown angle between the external line and the circle.
- Using the exterior angle theorem in circles, which states that the exterior angle is equal to the opposite interior angle, we can write:
\[
b = \angle XYZ = 30^\circ.
\]

### Summary:
- In figure (a), \( \angle XOZ = 140^\circ \).
- In figure (b), \( a = 1.8 \, \text{cm} \).
- In figure (c), \( b = 30^\circ \).

Would you like more details or further explanations? Here are five related questions for further practice:

1. How would the solution change if the angle \( \angle XYZ \) in figure (a) was different?
2. How can you apply the intersecting chords theorem to other cases with more complex figures?
3. What is the general formula for angles formed by intersecting chords in a circle?
4. How would you find the area of the triangle formed by the chords in figure (b)?
5. What other circle theorems apply to figure (c)?

**Tip**: For geometry problems, always check if basic circle theorems apply (e.g., inscribed angles, central angles, or intersecting chords) to simplify the solution.

Find a and b in the following figures (a, b, and c), involving circle geometry and angles.

This solution covers key circle geometry concepts such as the angle at the center theorem, intersecting chords theorem, and exterior angle theorem. Suitable for high school students (Grades 9-12), these problems involve finding unknown angles and chord lengths.

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