Find the equation of the perpendicular bisector of AB.

Understand the Problem

The question asks for the equation of the perpendicular bisector of a line segment AB. To find this, we need the midpoint of segment AB and the slope of the line perpendicular to AB.

Answer

$$ y - \frac{y_1 + y_2}{2} = -\frac{1}{\frac{y_2 - y_1}{x_2 - x_1}} \left(x - \frac{x_1 + x_2}{2}\right) $$
Answer for screen readers

The final equation of the perpendicular bisector of line segment AB, using the above formulas, is:

$$ y - \frac{y_1 + y_2}{2} = -\frac{1}{\frac{y_2 - y_1}{x_2 - x_1}} \left(x - \frac{x_1 + x_2}{2}\right) $$

Steps to Solve

  1. Find the midpoint of segment AB

To find the midpoint ( M ) of a line segment AB with endpoints ( A(x_1, y_1) ) and ( B(x_2, y_2) ), use the midpoint formula:

$$ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) $$

  1. Calculate the slope of line AB

To find the slope ( m ) of line segment AB, use the slope formula:

$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$

  1. Determine the slope of the perpendicular line

The slope of the perpendicular bisector ( m_{\perp} ) is the negative reciprocal of the slope of line AB. Thus:

$$ m_{\perp} = -\frac{1}{m} $$

  1. Write the equation of the perpendicular bisector

Using point-slope form, the equation of the line can be written as:

$$ y - y_0 = m_{\perp}(x - x_0) $$

Where ( (x_0, y_0) ) is the midpoint ( M ).

The final equation of the perpendicular bisector of line segment AB, using the above formulas, is:

$$ y - \frac{y_1 + y_2}{2} = -\frac{1}{\frac{y_2 - y_1}{x_2 - x_1}} \left(x - \frac{x_1 + x_2}{2}\right) $$

More Information

The perpendicular bisector of a line segment has some interesting properties: it divides the segment into two equal parts and every point on the bisector is equidistant from the endpoints A and B. This is useful in geometry and in constructions, such as finding circumcenters of triangles.

Tips

  • Forgetting to compute the negative reciprocal for the slope of the perpendicular line.
  • Miscalculating the midpoint coordinates.
  • Using the wrong formula for the slope.

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