point A(-1,-2) B(7,2) C(k,4) K is a constant. The vertices of ABC. Angle ABC is a right angle.
Ive worked out the gradient of AB to be 1/2. And the negative reciprocals is 2.
How do you calculate the value of k??
thanks for the help
To calculate the value of k, we need to use the fact that angle ABC is a right angle. In a right-angled triangle, the product of the gradients of two perpendicular lines is always -1.
The gradient of AB is given as 1/2. Therefore, the gradient of the line joining points A and C must be -2 (the negative reciprocal of 1/2).
Let's find the gradient of the line joining A and C using the formula for finding the gradient between two points.
Using the formula for the gradient:
m = (y2 - y1) / (x2 - x1)
Substituting the values for points A and C:
-2 = (4 - (-2)) / (k - (-1))
Simplifying:
-2 = 6 / (k + 1)
Cross-multiplying:
-2(k + 1) = 6
Expanding:
-2k - 2 = 6
Simplifying further:
-2k = 8
Dividing by -2:
k = -4
Therefore, the value of k is -4.