Question

Points A(11,12), E(6,−11), and D(−4,−3) are given. Line AE intersects line AB at point C(6,k). Determine the value of k.

a) -3
b) 6
c) 8
d) -2

Answer 1

To determine the value of k, we need to find the equation of line AB and then find the coordinates of point C where line AE intersects line AB. Using the slope-intercept form, we can find the equation of line AB. Substituting the x-coordinate of point C into the equation of line AB, we can solve for k.

Step-by-step explanation:

To determine the value of k, we need to find the equation of line AB and then find the coordinates of point C where line AE intersects line AB.

First, let's find the equation of line AB. We can use the slope-intercept form, which is y = mx + b.

Given points A(11,12) and E(6,-11), we can find the slope (m) using the formula m = (y2 - y1) / (x2 - x1). Plugging in the values, we get m = (-11 - 12) / (6 - 11) = -23/5. Now we can substitute the slope and any coordinate (x, y) of point A into the slope-intercept form to find the y-intercept (b). Using A(11,12) as the coordinate, we get 12 = (-23/5)(11) + b. Solving for b, we find b = 257/5.

Now we can find the equation of line AB: y = (-23/5)x + 257/5. Next, we find the coordinates of point C where line AE intersects line AB. Since C(6,k), we substitute x = 6 into the equation of line AB to solve for k. Using the equation y = (-23/5)x + 257/5, we get k = (-23/5)(6) + 257/5 = -138/5 + 257/5 = 119/5. Therefore, the value of k is 119/5 or 23.8.