https://file.aiquickdraw.com/mathbot/file/1726407560142k5oyxg1o.jpg

In the provided image, we have a triangle with vertices at points \( A(-2,1) \), \( B(3,1) \), and \( C(4,-2) \). The question asks for the height relative to side \( AB \).

### Step 1: Understanding Side \( AB \)
The coordinates of \( A \) and \( B \) are \( A(-2, 1) \) and \( B(3, 1) \). Since both points have the same \( y \)-coordinate (1), this indicates that side \( AB \) is a horizontal line.

#### Length of Side \( AB \)
To calculate the length of side \( AB \), we can use the distance formula for two points \( (x_1, y_1) \) and \( (x_2, y_2) \):
\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Since the points are \( A(-2,1) \) and \( B(3,1) \), we substitute:
\[
\text{AB} = \sqrt{(3 - (-2))^2 + (1 - 1)^2} = \sqrt{(3 + 2)^2} = \sqrt{5^2} = 5
\]
Thus, the length of side \( AB \) is 5 units.

### Step 2: Height Relative to \( AB \)
The height relative to side \( AB \) is the perpendicular distance from point \( C(4,-2) \) to the line \( AB \). Since side \( AB \) is horizontal at \( y = 1 \), the height will be the vertical distance between the \( y \)-coordinate of \( C \) and \( y = 1 \).

Thus, the height \( h \) is:
\[
h = |y_C - y_{AB}| = |-2 - 1| = 3
\]

### Conclusion
The height relative to side \( AB \) measures **3** units. Therefore, the correct answer is \( \boxed{3} \).

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Would you like further details or have any additional questions?

Here are 5 related questions to expand your understanding:
1. How do you find the slope of a line between two points?
2. What is the general formula for the distance between two points?
3. How do you find the area of a triangle using base and height?
4. What is the equation of a line parallel to a given line?
5. How do you determine the midpoint of a line segment?

**Tip:** Always check if two points share the same \( x \)- or \( y \)-coordinate when determining if a line is vertical or horizontal.

Discover how to determine the height relative to side AB in a triangle with vertices at A(-2,1), B(3,1), and C(4,-2). This guide explains using the distance formula and absolute value, suitable for students in Grades 8-10.

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