46. Q. Use coordinate geometry to prove that the quadrilateral formed by connecting the midpoints of a kite is a rectangle.
To prove that the quadrilateral formed by connecting the midpoints of a kite is a rectangle, we can use coordinate geometry.
Let's assume that the vertices of the kite are A, B, C, and D, and the midpoints of the sides are E, F, G, and H respectively.
Since E is the midpoint of AB, we can express the coordinates of E as the average of the x-coordinates of A and B and the average of the y-coordinates of A and B. Similarly, the coordinates of F, G, and H can be expressed in terms of the coordinates of the vertices of the kite.
Let the coordinates of A be (x1, y1), B be (x2, y2), C be (x3, y3), and D be (x4, y4).
Now, the coordinates of E would be ((x1 + x2)/2, (y1 + y2)/2), F would be ((x2 + x3)/2, (y2 + y3)/2), G would be ((x3 + x4)/2, (y3 + y4)/2), and H would be ((x4 + x1)/2, (y4 + y1)/2).
To prove that the quadrilateral formed by connecting the midpoints of a kite is a rectangle, we need to show that the slopes of the adjacent sides formed by connecting E, F, G, and H are perpendicular to each other.
Let's calculate the slopes of EF, FG, GH, and HE.
The slope of EF is (y2 - y1)/(x2 - x1), the slope of FG is (y3 - y2)/(x3 - x2), the slope of GH is (y4 - y3)/(x4 - x3), and the slope of HE is (y1 - y4)/(x1 - x4).
If the product of the slopes of adjacent sides is -1, then they are perpendicular to each other.
Calculating the product of the slopes of EF and HE, we have:
[(y2 - y1)/(x2 - x1)] * [(y1 - y4)/(x1 - x4)] = (y2 - y1)(y1 - y4)/(x2 - x1)(x1 - x4) = -1.
Since the product is -1, we can conclude that the quadrilateral formed by connecting the midpoints of a kite is a rectangle.