In the x-y coordinate axis, a perpendicular line AD is drawn from the

Question

In the x-y coordinate axis, a perpendicular line AD is drawn from the point A (2,4) on the line x + y = 10 and it is extended till E, such that AD = DE. Find the co-ordinate of E?

A. (-6,-8)
B. (55/13, 44/13)
C. (6, 8)
D. (11, 5)
E. (15,15)

LakshitPareek · Answer

Hey, can someone please show me, how this will be solved by the mid point formula (m1x1 + m2x2/m1+m2, m1y1 + m2y2/m1+m2)
i tried it, but it is not giving me the right answer. i want to know where i am going wrong.
thanks in advance

Suhasi · Answer

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

GynCliff · Answer

Like I always say GMAT will punish you if you're a stickler of methods and procedures.

In this question the trick is to understand the question and take a glance of the answers so that you're able to know how far to solve or how to tackle.

One, the line is perpendicular to the other because it's mention and also they meet. So in this case, they're are testing on the relationship of gradients of perpendicular lines. That is the product of perpendicular lines is -1.

So in this question is will use the answers and calculate backwards to find a coordinate that when used together with (2,4), they give you a gradient that is multiplied by the gradient you get from rearranging the other line's equation you get -1.

The equation y + x = 10 in terms of y = mx + c

y = -x + 10

meaning the gradient is -1

so the gradient of the other line if multiplied by -1 must be equal to -1

From this point you pick any of the answer choices and try to calculate the gradient and any answer that gives you a gradient of 1 that is the answer.

In this case ANSWER C is the correct one because

(6,8) and (2, 4) will give you a gradient of 1

And 1 times -1 = -1

CORRECT ANSWER C

ScottTargetTestPrep · Answer

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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In the x-y coordinate axis, a perpendicular line AD is drawn from the

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