Question

Даны точки B (-2;10) C(6;4) D(1;1) и точка А лежащая на оси OY найти координаты точки А если прямая АС перпендикулярна прямой BD?

Answer 1

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To find the coordinates of point A, we need to use the fact that the line AC is perpendicular to the line BD. This means that the slopes of these lines are opposite reciprocals, that is, k1 * k2 = -1, where k1 is the slope of AC and k2 is the slope of BD.

To find the slope of a line, we can use the formula k = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are any two points on the line. For example, to find the slope of BD, we can use the points B (-2, 10) and D (1, 1) and plug them into the formula:

k2 = (y2 - y1) / (x2 - x1) = (1 - 10) / (1 - (-2)) = -9 / 3 = -3

Similarly, to find the slope of AC, we can use the point C (6, 4) and the point A (0, y), where y is the unknown coordinate of A. Note that the x-coordinate of A is 0 because A lies on the y-axis. Plugging these points into the formula, we get:

k1 = (y2 - y1) / (x2 - x1) = (y - 4) / (0 - 6) = (y - 4) / (-6)

Now, using the fact that k1 * k2 = -1, we can solve for y:

k1 * k2 = -1 (y - 4) / (-6) * (-3) = -1 (y - 4) / 2 = -1 y - 4 = -2 y = 2

Therefore, the coordinates of point A are (0, 2).