ABCD is a quadrilateralwith vertices A(2;5),B(-3;10),C(-4;3) and D(1;-2).then show that BD and AC bicect each at right angles at M.
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To show that BD and AC bisect each other at right angles at point M, we need to demonstrate that the two line segments BD and AC intersect at a point, and that the angles formed at that point are right angles.
Step 1: Find the equations of the lines BD and AC
- Line BD passes through points B(-3,10) and D(1,-2). We can find the equation of this line using the point-slope form: y - y1 = m(x - x1), where (x1, y1) is a point on the line, and m is the slope. Calculate the slope of BD using the formula: m = (y2 - y1) / (x2 - x1), where (x2, y2) are the coordinates of a second point on the line.
- Line AC passes through points A(2,5) and C(-4,3). Again, use the point-slope form to find its equation.
Step 2: Find the intersection point
- Set the equations of BD and AC equal to each other and solve for x and y. The resulting values will be the coordinates of the point of intersection, M.
Step 3: Calculate the slopes of BM and AM
- Use the coordinates of B and M to find the slope of line BM.
- Use the coordinates of A and M to find the slope of line AM.
Step 4: Check if the slopes of BM and AM are negative reciprocals
- If the slopes of BM and AM are negative reciprocals (meaning their product is -1), it shows that the two lines are perpendicular and intersect at right angles.
If all these steps are followed correctly and the slopes of BM and AM turn out to be negative reciprocals, you will have proven that BD and AC bisect each other at right angles at point M.