https://file.aiquickdraw.com/mathbot/file/1727366348751a1crdwqo.jpg

The problem asks to determine the equation of the perpendicular bisector of a segment with endpoints \( A(0, 4) \) and \( B(4, 2) \). It also requires finding the coordinates of the midpoint of segment \( AB \) and verifying that the midpoint lies on the perpendicular bisector.

### Steps to solve:

1. **Midpoint of segment \( AB \):**
   The midpoint \( M \) of a segment with endpoints \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by:
   \[
   M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
   \]
   Substituting the coordinates of points \( A(0, 4) \) and \( B(4, 2) \):
   \[
   M = \left( \frac{0 + 4}{2}, \frac{4 + 2}{2} \right) = (2, 3)
   \]
   So, the midpoint of the segment \( AB \) is \( M(2, 3) \).

2. **Slope of segment \( AB \):**
   The slope \( m \) of a line passing through two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by:
   \[
   m = \frac{y_2 - y_1}{x_2 - x_1}
   \]
   Substituting the coordinates of \( A(0, 4) \) and \( B(4, 2) \):
   \[
   m = \frac{2 - 4}{4 - 0} = \frac{-2}{4} = -\frac{1}{2}
   \]
   The slope of line \( AB \) is \( -\frac{1}{2} \).

3. **Slope of the perpendicular bisector:**
   The perpendicular bisector has a slope that is the negative reciprocal of the slope of line \( AB \). Since the slope of \( AB \) is \( -\frac{1}{2} \), the slope of the perpendicular bisector is:
   \[
   m_{\text{bisector}} = 2
   \]

4. **Equation of the perpendicular bisector:**
   The equation of a line with slope \( m \) passing through a point \( (x_1, y_1) \) is given by:
   \[
   y - y_1 = m(x - x_1)
   \]
   Substituting the slope \( 2 \) and the coordinates of the midpoint \( M(2, 3) \):
   \[
   y - 3 = 2(x - 2)
   \]
   Simplifying:
   \[
   y - 3 = 2x - 4 \implies y = 2x - 1
   \]
   So, the equation of the perpendicular bisector is \( y = 2x - 1 \).

5. **Verification:**
   We already found that the midpoint \( M(2, 3) \) has coordinates \( (2, 3) \). Substituting \( x = 2 \) into the equation of the perpendicular bisector:
   \[
   y = 2(2) - 1 = 4 - 1 = 3
   \]
   Since the point \( (2, 3) \) satisfies the equation \( y = 2x - 1 \), we verify that the midpoint lies on the perpendicular bisector.

### Final Answer:
- The equation of the perpendicular bisector is \( y = 2x - 1 \).
- The midpoint of segment \( AB \) is \( M(2, 3) \), and it lies on the perpendicular bisector.

Would you like further clarification or details?

Here are 5 related questions to practice:

1. How would you find the perpendicular bisector of a vertical segment?
2. What is the geometric significance of the perpendicular bisector?
3. Can the perpendicular bisector intersect the segment at a point other than the midpoint?
4. How would the equation change if the segment had endpoints \( A(-3, 7) \) and \( B(5, -1) \)?
5. How would the equation be different if the slope of segment \( AB \) were positive?

**Tip:** The slope of a perpendicular bisector is always the negative reciprocal of the slope of the original line.

Individua l'equazione dell'asse del segmento di estremi A(0, 4) e B(4, 2) utilizzando la definizione di asse come luogo dei punti equidistanti da A e B. Determina infine le coordinate del punto medio del segmento AB e verifica che appartiene all'asse.

Find the equation of the perpendicular bisector of the segment with endpoints A(0, 4) and B(4, 2). Learn how to calculate the midpoint and verify that it lies on the bisector. Suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation