https://file.aiquickdraw.com/mathbot/file/1726775672086zahrglnh.jpg

Let's analyze the diagram and the accompanying questions. Here's the breakdown of the figure and the true statements to fill in:

### **Key Observations from the Figure:**
- Points \( N \), \( K \), \( J \), and \( H \) lie on plane \( Z \).
- Points \( L \) and \( M \) do not lie on plane \( Z \) (as shown by the dashed line indicating they are outside the plane).
- \( \overline{ML} \) is a line outside of plane \( Z \), passing through points \( L \) and \( M \).
- The diagram shows a vertical line in the plane, which is not labeled, but it's reasonable to assume it's \( \overline{NJ} \) or \( \overline{KJ} \).

### **Solutions:**

1. **(a)** Another name for plane \( Z \) is plane **\( HJK \)**.
   - Since any three non-collinear points can define a plane, and points \( H \), \( J \), and \( K \) lie in plane \( Z \), we can name the plane as plane \( HJK \).

2. **(b)** \( H \), \( N \), \( K \), and **\( J \)** are distinct points that are coplanar.
   - These four points are all in the same plane \( Z \), so they are coplanar.

3. **(c)** \( N \) and \( K \) are distinct points that are collinear.
   - Points \( N \) and \( K \) lie on the same line in the plane, as indicated by the figure.

4. **(d)** \( \overline{ML} \) and **\( \overline{NJ} \)** are distinct lines that intersect.
   - The line \( \overline{ML} \) (which passes through points \( L \) and \( M \)) intersects line \( \overline{NJ} \) (which lies in the plane and passes through points \( N \) and \( J \)) at point \( J \).

---

### Questions for further practice:

1. How do you define a plane in geometry?
2. Can you list more points that are coplanar in the figure?
3. What is the difference between collinear and coplanar points?
4. How can two distinct lines intersect in space?
5. How would the problem change if point \( M \) was also in plane \( Z \)?

### Tip:
When determining whether points are coplanar, check if all points lie on the same flat surface or plane. Three non-collinear points always define a plane.

In the figure below, points N, K, J, and H lie in plane Z. Points L and M do not lie in plane Z. For each part below, fill in the blanks to write a true statement.

Explore how points N, K, J, and H define plane Z and interact with other elements outside the plane such as L and M. This detailed explanation helps visualize planes, collinear and coplanar points in geometry, suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation