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
- 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:
$$ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) $$
- Calculate the slope of line AB
To find the slope ( m ) of line segment AB, use the slope formula:
$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$
- 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:
$$ m_{\perp} = -\frac{1}{m} $$
- 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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