What is the shortest distance between the following 2 lines: x + y = 3

Question

What is the shortest distance between the following 2 lines: x + y = 3 and 2x + 2y = 8?

(A) 0
(B) 1/4
(C) 1/2
(D)  \frac{{\sqrt 2}}{2} 
(E)  \frac{{\sqrt 2}}{4}

Skywalker18 · Accepted Answer

The shortest distance between any two lines on the xy axis would be either 
0 - the lines intersect 
or the lines are parallel.

The two lines are:
x + y = 3
2x + 2y = 8
=>x + y = 4
The given 2 lines are parallel. (a1/a2 = b1/b2 )

The lines intersect x and y axis - (3,0) , (4,0) , (0,3) and (0,4)

In  	riangle  ABC 
ac= 1 ,
 \angle  bca =45 degrees 
so it is a 45-45-90 triangle. Since ac = 1 ,
Sin 45 = ab/ac = ab/1 = 
=1/2
=> ab =1/(2)^(1/2)
The sides in a 45-45-90 are 1:1:2√2

The distance between the 2 lines = ab = 1/(2)^(1/2) or (2)^(1/2) / 2

Answer D

robu · Answer

first write the equations in the form:

x+y-3=0
x+y-4=0

Now the formula for shortest/ perpendicular distance = Mod(C1-C2)/sqrt (a^2+b^2)
i.e.
Mode (C1-C2) = mode (-3-(-4)) =1
Sqrt(a^2+b^2)= 1^2+1^2 =root 2.

answer= 1/sqrt2= sqrt2/2.

I hope it makes some sense!

Ekland · Answer

The explanation above seems like magic.
Can someone offer a learnable explanation?

Skywalker18 · Answer

Hi Nez , I did not intend to give a magical explanation

. Let me specify the steps that were followed - The steps that i have followed are - 1. The given lines are parallel 2. Find where the given 2 lines intersect with X and Y axis . The line x+y=3 (-- line 1) will pass through (3,0) and (0,3) and the line x + y = 4 (-- line 2) will pass through (4,0) and (0,4) 3. The shortest distance between 2 parallel lines is the perpendicular distance. So , a perpendicular- ab has been dropped from line 2 to line 1. In the riangle which has vertices at (0,0) , (4,0) and (0,4) We have an isosceles right riangle => \angle bca =45 degrees Coordinates of point a are (3,0) and point c are (4,0) Hope it helps!!

PMZ21 · Answer

Two lines:

L1: x + y = 3 --> y = (-1)x + 3, where slope m1 = -1 and b = 3
L2: 2x + 2y = 8 --> 2y = -2x + 8 --> y = (-1) + 4, where slope m2 = -1 and c = 4

Since the two slopes are equal, line 1 is parallel to line 2.

Shorted distance between two parallel lines can be found by using the formula: | b-c | / (m^2 + 1)^1/2
==> |4-3| /  ^1/2
==> 1/ (2)^1/2
==> Option D

sjavvadi · Answer

I like the 45-45-90 solution because it explains the solution conceptually.

Just in case as an alternate method I have attached the solution using the the "mid-point" information and then the distance between two points.

Shruti0805 · Answer

Lets consider the points where Line 1 and Line 2 intersect the X and Y axes:

L1 : x + y = 3
=> y= -x + 3
Substitue x=0 to get y=3 and y=0 to get x=3.
So, the 2 points are (0,3) and (3,0).

Similarly for L2 : 2x+2y = 8
=> y = -x + 4
Points are (0,4) and (4,0).

Check, in the attached figure, both the lines are parallel to each other and create a right triangle with the X and Y axes.
Now, in right triangles, we know that Base*Height = Hypotenuse*Perpendicular dropped from the 90 degrees vertex.
Also notice because the 2 shorter sides are equal, that makes it a 45-45-90 triangle, thereby, the sides would be in the ratio  1:1:\sqrt{2}

So, for the 1st triangle, using the distance formula for both the smaller sides,  3*3 = 3\sqrt{2} * X 
=>  X = \frac{3}{\sqrt{2}}

Similarly, for the 2nd triangle,  4*4 = 4\sqrt{2} * Y 
=> Y =  \frac{4}{\sqrt{2}}

Shortest distance is always the perpendicular distance between 2 lines, if they're parallel, so in our case, the distance should be (Y-X)
=  \frac{1}{\sqrt{2}} 
Option D.

atomicmass · Answer

Answer D

Solution 
X + Y = 3 -------------------------I
X + Y = 4 -------------------------II

Slope of both the above lines is -1. therefore, both the lines will make angle of 45 with the X coordinate and Y coordinate.

Line perpendicular both of the above lines must have slope 1.
Line Y = X has slope 1.

Substitute the Y = X and point of lines passing at 90 degree.
I -> x = 1.5, y = 1.5 ------------------------------III
II -> x = 2, y = 2 -----------------------------------IV

Using distance formula on III and IV we get  \frac{1}{{\sqrt 2}} 
or
 \frac{{\sqrt 2}}{2}

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

What is the shortest distance between the following 2 lines: x + y = 3

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

What is the shortest distance between the following 2 lines: x + y = 3

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards