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We are given a right triangle \( \triangle ABC \) with the right angle at \( C \). The coordinates of points \( A(0, 2) \), \( C(1, 0) \), and the horizontal coordinate of point \( B(4, y) \) are provided, and we need to find the vertical coordinate \( y \) of point \( B \).

### Steps to solve:

1. **Distance between points**: Since \( \triangle ABC \) is a right triangle, we can apply the Pythagorean theorem to the distances between points \( A \), \( B \), and \( C \). The distances from \( A \) to \( C \) and from \( B \) to \( C \) form the legs of the right triangle, while the distance from \( A \) to \( B \) is the hypotenuse.

2. **Calculate distances**:
   - The distance from \( A(0, 2) \) to \( C(1, 0) \) is:
     \[
     AC = \sqrt{(1-0)^2 + (0-2)^2} = \sqrt{1^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5}
     \]
   - The distance from \( C(1, 0) \) to \( B(4, y) \) is:
     \[
     CB = \sqrt{(4-1)^2 + (y-0)^2} = \sqrt{3^2 + y^2} = \sqrt{9 + y^2}
     \]
   - The distance from \( A(0, 2) \) to \( B(4, y) \) is:
     \[
     AB = \sqrt{(4-0)^2 + (y-2)^2} = \sqrt{4^2 + (y-2)^2} = \sqrt{16 + (y-2)^2}
     \]

3. **Apply the Pythagorean theorem**: 
   Since \( AC^2 + BC^2 = AB^2 \), we substitute the distances:
   \[
   (\sqrt{5})^2 + (\sqrt{9 + y^2})^2 = (\sqrt{16 + (y-2)^2})^2
   \]
   Simplifying the equation:
   \[
   5 + (9 + y^2) = 16 + (y-2)^2
   \]
   Expanding the right-hand side:
   \[
   5 + 9 + y^2 = 16 + y^2 - 4y + 4
   \]
   Simplify further:
   \[
   14 + y^2 = 20 + y^2 - 4y
   \]
   Cancel \( y^2 \) from both sides:
   \[
   14 = 20 - 4y
   \]
   Solve for \( y \):
   \[
   -4y = 14 - 20 = -6 \quad \Rightarrow \quad y = \frac{6}{4} = 1.5
   \]

Thus, the vertical coordinate of point \( B \) is \( y = 1.5 \).

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. How would the solution change if \( C \) were located at a different point?
2. What other methods could you use to find the vertical coordinate of point \( B \)?
3. How does the Pythagorean theorem apply to triangles in coordinate geometry?
4. How would you solve the problem if given the length of the hypotenuse?
5. How can you determine if a triangle is a right triangle using only its vertex coordinates?

**Tip**: When working with right triangles on the coordinate plane, always remember to check if the right angle is at the expected vertex using the Pythagorean theorem or slope analysis.

As shown below, in triangle ABC, angle ACB = 90°. If we know the coordinates of points A(0,2), C(1,0), and the horizontal coordinate of point B is 4, then the vertical coordinate of point B is ( ).

Learn how to calculate the vertical coordinate of point B in a right triangle on the coordinate plane using the Pythagorean theorem. The problem involves key concepts of distance formula and coordinate geometry, suitable for students in Grades 9-11.

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