To find the length of line segment AB, where A and B are midpoints of segments XY and XZ respectively, we need the coordinates of points X, Y, and Z.

Assuming the coordinates are given as follows:

• X(x₁, y₁)
• Y(x₂, y₂)
• Z(x₃, y₃)

Steps to Solve:

1. Calculate the Midpoints:

   • For midpoint A of XY: $A\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$
   • For midpoint B of XZ: $B\left(\frac{x_1 + x_3}{2}, \frac{y_1 + y_3}{2}\right)$

2. Find the Length of AB: Use the distance formula: $AB = \sqrt{\left(\frac{x_1 + x_3}{2} - \frac{x_1 + x_2}{2}\right)^2 + \left(\frac{y_1 + y_3}{2} - \frac{y_1 + y_2}{2}\right)^2}$ Simplifying gives: $AB = \sqrt{\left(\frac{x_3 - x_2}{2}\right)^2 + \left(\frac{y_3 - y_2}{2}\right)^2}$ $AB = \frac{1}{2}\sqrt{(x_3 - x_2)^2 + (y_3 - y_2)^2}$

3. Plug in the Coordinates: Substitute the coordinates of points Y and Z into the formula to find the length of AB.

4. Round to the Nearest Tenth: After calculating the value, round it to the nearest tenth.

Example Calculation:

Let's say:

• X(2, 3)
• Y(4, 7)
• Z(6, 5)

1. Calculate Midpoints: $A\left(\frac{2 + 4}{2}, \frac{3 + 7}{2}\right) = (3, 5)$ $B\left(\frac{2 + 6}{2}, \frac{3 + 5}{2}\right) = (4, 4)$

2. Calculate AB: $AB = \sqrt{(4 - 3)^2 + (4 - 5)^2} = \sqrt{1 + 1} = \sqrt{2} \approx 1.4$

Final Answer: The length of AB is approximately 1.4.

Questions:

1. What are the coordinates of points X, Y, and Z?
2. How do midpoints affect the geometry of a triangle?
3. Can this method be used for 3D coordinates as well?
4. What is the significance of midpoints in geometric constructions?
5. How would the length of AB change if the coordinates of Y or Z changed?

Tip: Always visualize the points on a coordinate plane to better understand the relationships between them.