Suhasi wrote:
Bunuel ScottTargetTestPrep Can you please help with this question? I'm unable to understand the approach needed to solve it. I understand that the lines being perpendicular will have negative reciprocal slopes but I can't seem to get any further. Thanks in advance
I think the easiest way to solve this question is indeed using the fact that the product of the slopes of perpendicular lines is -1.
We are told that a perpendicular is drawn from the point (2, 4) onto the line x + y = 10. Let's calculate the slope of this perpendicular. Rewriting the line x + y = 10 in the slope-intercept form as y = -x + 10, we see that this line has a slope of -1. Therefore, the perpendicular to this line has a slope of 1.
Next, let's write the equation of this perpendicular using the slope and the point (2, 4) that we are told to be contained on this line. Since the slope is 1, the equation of the line will be of the form y = x + b for some b. To find b, let's substitute x = 2 and y = 4 in y = x + b:
4 = 2 + b
b = 2
Thus, the equation of the line which contains the point (2, 4) and which is perpendicular to x + y = 10 is y = x + 2.
Next, let's find the point of intersection of the lines y = -x + 10 and y = x + 2.
x + 2 = -x + 10
2x = 8
x = 4
Substituting x = 4 in y = x + 2, we find y = 6. Therefore, the lines y = -x + 10 and y = x + 2 intersect at the point (4, 6).
We are ready to calculate the coordinates of point E. Let E = (a, b). Notice that the point (4, 6) must be the midpoint of the points (2, 4) and (a, b). Using the midpoint formula, we obtain:
((2 + a)/2, (4 + b)/2) = (4, 6)
Solving (2 + a)/2 = 4, we find that a = 6. Since there is only one answer choice with an x-coordinate of 6, the answer must be C. However, we can verify that the y-coordinate of E is 8. We can either solve (4 + b)/2 = 6, or substitute x = 6 in y = x + 2. In either case, we'll obtain b = 8.
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