HOT Competition (Instagram) 27 Aug/11:45PM: In the rectangular coordi

Question

In the rectangular coordinate plane, if line segment AB has a length of 10, what are the coordinates of the midpoint of line segment AB?

(1) the slope of line segment AB is -3 
(2) the coordinates of point A are (2,6)

M36-03

Bunuel · Accepted Answer

Official Solution:

In the rectangular coordinate plane, if line segment AB has a length of 10, what are the coordinates of the midpoint of line segment AB? 
 
(1) The slope of line segment AB is -3. We know only the slope and the length of AB but not the coordinates of the points. Not sufficient.
 
(2) The coordinates of point A are (2,6). We know nothing about point B. Not sufficient.
 
(1)+(2) What do we know? The length of AB, the slope of AB and the coordinates of A. To get the coordinates of the midpoint, we need also to know the exact position of point B. But B can be above point A as well as below point A, so two different positions of B are possible, giving two different coordinates of the midpoint of AB. Not sufficient.
 
For better understanding check the image below:
 
 https://gmatclub.com/tests-beta/resources/import/img/M36-03.png

Answer: E

sood1596 · Answer

I guess the answer is E

Given distance is 10
Using the distance formula

10= ((x2-x1)^2 + (y2-y1)^2)^1/2

1)
M= y2-y1/(x2-x1)
We need some values to plug in 
Not sufficient
Eliminate A and D

2) 
2,6 is A
Not sufficient we need B as well

Both together not sufficient

Answer E

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acerohit1 · Answer

Give in the question line segment = 10 
Ie √(x2-x1)^2 + (y2-y1)^2 = 10
 We need to find the coordinator 
Statement 1 y2-y1/x2-x1 = -3 
Not sufficient to answer 
Statement 2 a coordinate are 2 and 6 still not sufficient 
Adding 1 and 2 we get 
Y2-6/X2-2 = -3 
Y2 -6 = -3x2 + 6 
Y2 +3x2 = 12
Still we are not able. To get any value 
so insufficient 
Answer is e

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JonShukhrat · Answer

Attached

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Jks3000 · Answer

Statement 1 : the slope of line segment AB is -3

It is not possible to find the coordinates of the midpoint of line segment AB because we do not know either the coordinate of point A or point B.

Hence Insufficient

Statement 2  : the coordinates of point A are (2,6)

Coordinates of point A are given but we do not know the slope of line segment AB.

Hence Insufficient

Combining Statement 1 and 2 :

Now use little bit of logic here. ( Gmat is all about Logic !!)

You have coordinates of point A and a slope, But you don't know whether point B will lie " down the line of slope " OR " up the line of the slope ". Hence from the given information you can get 2 valid values of midpoint of line AB.( depending on the position of B )

Hence Insufficient.

Correct answer is Option E. Neither.

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rajatchopra1994 · Answer

Explanation:

Given: AB has a length of 10

To find: co-ordinates of midpoint of line segment AB

Point A(X1,Y1) 
Point B(X2,Y2)
Mid points: (X1+X2)/2 ,(Y1+Y2)/2

Statement 1: the slope of line segment AB is -3

-3 = (Y2-Y1)/(X2-X1)

We don't know any cordinations value of point A & B

Not Sufficient

Statement 2: the coordinates of point A are (2,6)
X1=2; Y1=6
Length = 10 
Length = √{(X2-X1)^2 + (Y2-Y1)^2}
By putting values of X1; Y1 and length we get below equation.
60 = (X2)^2 + (Y2)^2 - 4(X2) - 12(Y2) --- 1

Not possible to solve the above eqn as 2 variables.

Not Sufficient.

From Statement 1 & 2; 
-3 = (Y2-Y1)/(X2-X1)
-3 = (Y2-6)/(X2-2)
12 = 3(X2) + (Y2)----2

From Eqn 1 & 2; we will get 2 values each for X2 & Y2.

So 2 mid point's.

As line segment can be drawn in any of the 2 direction from Point A. Also attaching graph image for reference.

Not Sufficient.

IMO-E

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MayankSingh · Answer

IMO E

Given :   Length of AB=10  
 Check:  Coordinates of the midpoint of line segment AB?

If a line segment has end points coordinate as A (a,b) & B(c,d) then the midpoint of line segment AB is (a+c)/2, (b+d)/2 
Length^2 = 10^2 = (a-c)^2 + (b-d)^2

(1)   the slope of line segment AB is -3  
 Let Coordinate be A (a,b) & B(c,d)
A/Q, (d-b)/(c-a) = -3
 d-b = -3(c-a)
 
 (a-c)^2 + (b-d)^2=10^2 
=> (a-c)^2 +9(a-c)^2 =10^2
=> 10 (a-c)^2 = 10^2
=> (a-c)^2 = 10
=>   (a-c) = ± √ 10 & (b-d)= ∓ 3√ 10  
But from this we can't find the values of coordinates. 
  Insufficient

(2)   the coordinates of point A are (2,6)  
 A (a,b)= 2,6
 (a-c)^2 + (b-d)^2=10^2 
 (2-c)^2 + (6-d)^2 =10^2 
Since we have no constrains on c,d , multiple values of c,d are possible. 
  Insufficient

Together A & B  
  (a,b)= 2,6
(a-c) = ± √ 10 =>  c = 2 ∓ √ 10 
(b-d)= ∓ 3√ 10 =>  d= 6 ± 3√ 10

Midpoint (a+c)/2, (b+d)/2 =(4 ∓ √ 10)/2, (12 ± 3√ 10)/2 
Therefore two values possible. 
  Insufficient

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BenjaminS · Answer

I will try to solve this problem by equation.
From the question, the length of line AB = 10. What are the coordinates of midpoint of line AB?
1) the slope of line segment AB is -3
So y = mx+c
Knowing m = -3; y = -3x+c
Therefore, the coordinates of point A and point B are many values. 
 Insufficient.

2) the coordinates of point A are (2,6)
Knowing A = (2,6), B can be many values that have the distance, which is 10.
 Insufficient.

1)+2): y = -3x+c and point A = (2,6) 
So the coordinates of point B can be two value. 
As a result, the midpoint of line AB cannot be found.
 Insufficient.

I choose E.

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MrBaby · Answer

In the rectangular coordinate plane, if line segment AB has a length of 10, what are the coordinates of the midpoint of line segment AB?

(1) the slope of line segment AB is -3
   According to attached file;  
 We know the blue line with its slope is -3. Know nothing about pointA and pointB.
  INSUFFICIENT.

(2) the coordinates of point A are (2,6)
 We know that A (2,6), but still know nothing about pointB.
  INSUFFICIENT.

Combine (1)&(2);
   According to attached file;  
 pointB can locate upper or lower pointA, so the coordinates of the midpoint of line segment AB can locate upper or lower pointA too. 2 possible points. 
 INSUFFICIENT.

ANS E.

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Urvashij · Answer

100= (x2-x1)^2+(y2-y1)^2

Y2-y1= -3(x2-x1)

Using both ens

100= (x2-x1)^2+ 9(x2-x1)^2

10=x2-x1)^2

X1= (10)^1\2 +2

X1+x2)/2 = (10)^1\2 + 4)/2.

Similarly mid point of y2 and y1

My answer is c

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