https://file.aiquickdraw.com/mathbot/file/1726586121611u5yqzyju.jpg

Let's break down and solve the two parts of the problem in the image you uploaded.

### (f) Show that the points \( X(2, 3) \), \( Y(6, -3) \), and \( Z(4, 0) \) are collinear.

To prove that these points are collinear, we need to show that the slope between any two pairs of these points is the same.

1. **Slope of \( XY \)**:
   The formula for the slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
   \[
   m = \frac{y_2 - y_1}{x_2 - x_1}
   \]
   For points \( X(2, 3) \) and \( Y(6, -3) \):
   \[
   m_{XY} = \frac{-3 - 3}{6 - 2} = \frac{-6}{4} = -\frac{3}{2}
   \]

2. **Slope of \( YZ \)**:
   Now, for points \( Y(6, -3) \) and \( Z(4, 0) \):
   \[
   m_{YZ} = \frac{0 - (-3)}{4 - 6} = \frac{3}{-2} = -\frac{3}{2}
   \]

3. **Slope of \( XZ \)**:
   Lastly, for points \( X(2, 3) \) and \( Z(4, 0) \):
   \[
   m_{XZ} = \frac{0 - 3}{4 - 2} = \frac{-3}{2} = -\frac{3}{2}
   \]

Since the slopes \( m_{XY} = m_{YZ} = m_{XZ} = -\frac{3}{2} \), the points are collinear.

### (g) Determine the value of \( x \) in the circle problem.

In the given diagram, point \( O \) is the center of the circle, and we are given:

- \( \angle AOC = 120^\circ \)
- \( \angle ABC = 2x - 4^\circ \)

Since \( \angle AOC \) is the central angle and subtends arc \( AC \), and \( \angle ABC \) is an inscribed angle subtended by the same arc, we know that the inscribed angle is half the measure of the central angle.

Thus, the relationship is:
\[
\angle ABC = \frac{1}{2} \times \angle AOC
\]
Substitute the known values:
\[
2x - 4 = \frac{1}{2} \times 120
\]
\[
2x - 4 = 60
\]
Solve for \( x \):
\[
2x = 60 + 4 = 64
\]
\[
x = \frac{64}{2} = 32
\]

Thus, the value of \( x \) is \( \boxed{32} \).

---

Would you like more details on any part of the solution?

Here are 5 related questions you might find interesting:
1. How do you find the area of a triangle given three points?
2. How do you determine if three points form a right triangle?
3. What is the difference between a central angle and an inscribed angle in a circle?
4. How can you find the radius of a circle given an angle and a chord?
5. How do you prove that four points form a cyclic quadrilateral?

**Tip:** For solving geometry problems involving circles, remember that inscribed angles are always half of the central angle that subtends the same arc.

Show that the points X(2, 3), Y(6, -3) and Z(4, 0) are collinear. Determine the value of x by solving the following problem with proper reasoning. In the given figure, consider O to be the center of the circle.

Solve a two-part geometry problem involving collinearity of three points (X, Y, Z) using the slope formula and determine the value of x using the inscribed angle theorem for a circle. The problem is suitable for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation