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.
How do you calculate the value of k??
thanks for the help
As BC is perpendicular to AB, being a right angled triangle. The gradient of BC would be -2.
For the equation of BC,
y-2=-2(x-7)
y-2=-2x+14
2x+y=16
Then put C(k,4) as x and y,
2(k)+(4)=16
2k+4=16
2k=16-4
2k=12
k=6
sorry but confused
confused about what ?
The property of slopes of perpendicular lines , or
the procedure to solve the equation ?
where you get slope of BC = 2/(k-4) from??
Well, calculating the value of k is like solving a little riddle. Since angle ABC is a right angle, the product of the gradients of AB and BC should be equal to -1.
Now, since the gradient of AB is 1/2, we can set up the equation: (1/2) * m(BC) = -1.
Solving this equation will give us the value of m(BC), which is the gradient of BC. And once we have that, we can figure out the value of k. It's like solving a puzzle, except with numbers!
Good luck cracking the code! And remember, if you need any more help, I'll be here to clown around with the answers.
To calculate the value of k in the given scenario, we can leverage the fact that angle ABC is a right angle.
First, let's calculate the gradient of BC. The gradient between two points (x1, y1) and (x2, y2) is given by the formula:
Gradient = (y2 - y1) / (x2 - x1)
In this case, the coordinates of B are (7, 2) and the coordinates of C are (k, 4). Thus, the gradient of BC is:
Gradient of BC = (4 - 2) / (k - 7) = 2 / (k - 7)
Now, since angle ABC is a right angle, the product of the gradients of AB and BC should be -1. In other words:
Product of gradients = (1/2) * (2 / (k - 7)) = -1
To solve this equation for k, we can cross multiply:
1 * (k - 7) = -2
k - 7 = -2
k = -2 + 7
k = 5
Hence, the value of k in this scenario is 5, which would make the coordinates of point C (5,4).
slope of AB = 1/2 , you had that
slope of BC = 2/(k-4)
since angle B is 90°, the above slopes must be negative reciprocals of each other, that is ....
2/(k-4) = -2
solve for k