how do write and graph a coordinate proof?

Isoceles triangle ABC with AB congruent to BC perpendicular bisector BD from B to AC

To write and graph a coordinate proof for an isosceles triangle, you can use the given information to find the coordinates of the vertices and then use distance and slope formulas to prove the desired properties.

Here's a step-by-step guide:

1. Start by assigning coordinates to the given points. Let's say A has the coordinates (x1, y1), B has the coordinates (x2, y2), and C has the coordinates (x3, y3).

2. Use the information that AB is congruent to BC to write an equation to find the coordinates of the points A and C. Since the triangle is isosceles, the x-coordinate of A should be equal to the x-coordinate of C. You can express this equation as: x1 = x3.

3. Next, find the midpoint of segment AC to locate point B. The midpoint formula states that the midpoint of a line segment with endpoints (x1, y1) and (x2, y2) is ((x1 + x2)/2, (y1 + y2)/2). Using this formula, find the coordinates of the midpoint of AC.

4. Now that you have the coordinates of points A, B, and C, you can calculate the slope of the line segment BD. The slope formula states that the slope of a line passing through two points (x1, y1) and (x2, y2) is (y2 - y1) / (x2 - x1). Calculate the slope of BD.

5. To find the slope of the perpendicular bisector BD, remember that the slopes of perpendicular lines are negative reciprocals of each other. Take the negative reciprocal of the slope of BD to find the slope of the perpendicular bisector.

6. Use the slope-intercept form of a line (y = mx + b) and substitute the coordinates of point B and the slope of the perpendicular bisector to find the equation of the line BD.

7. Finally, graph the triangle and the perpendicular bisector BD on a coordinate plane to visually represent the coordinate proof. Make sure that the lengths of AB and BC are equal and that BD forms a right angle with segment AC.

By following these steps, you can write and graph a coordinate proof for an isosceles triangle with a perpendicular bisector.