Final answer:
To find the coordinates of C, the point where the line AB cuts the y-axis, we can set the x-coordinate to zero. The coordinates of D, the midpoint of AB, can be found by taking the average of the x-coordinates and the average of the y-coordinates. The equation of the perpendicular bisector of AB can be found using the midpoint and the slope of AB. The coordinates of E, the point where the perpendicular bisector cuts the y-axis, can be found by substituting the y-coordinate as zero in the equation of the perpendicular bisector. To show that the area of triangle ABE is 4 times the area of triangle ECD, we can use the ratio of the lengths of their corresponding sides.
Step-by-step explanation:
Question:
Two Points A and B have coordinates (-3,2) and (9,8) respectively. Find the Coordinates of C, the point where the line AB cuts the y-axis.
- Coordinates of C: The line AB intersects the y-axis at C, where the x-coordinate is zero. Therefore, the coordinates of C is (0, yC).
- Coordinates of D: The midpoint of AB is found by taking the average of the x-coordinates and the average of the y-coordinates. So, the coordinates of D is [(xA + xB)/2, (yA + yB)/2].
- Equation of the Perpendicular Bisector: The perpendicular bisector of AB passes through the midpoint D and is perpendicular to AB. The equation of a line perpendicular to AB passing through D is given by y - yD = -1/slopeAB (x - xD).
- Coordinates of E: To find the coordinates of E, we substitute the y-coordinate as zero in the equation of the perpendicular bisector obtained in step (iii). Solving for x, we can find the x-coordinate of E. Therefore, the coordinates of E is (xE, 0).
- Area of Triangle ABE and ECD: To show that the area A of triangle ABE is 4 times the area of triangle ECD, we can use the fact that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding side lengths. Let's find the lengths of the sides of triangles ABE and ECD and compute their areas.