Distance b/w 2 lines

Question

What is the shortest distnace b/w the following 2 lines:
x+y=3 and
2x+2y=8

Well this seemed to be an easy one on the onset but i got it wrong..um sure i goofed up somewhere..

pls help me with the answer explanation. I will post the answer later.

KarishmaB · Answer

The shortest distance between any two lines on the xy axis would be either 0 (the lines will intersect somewhere) or the lines will be parallel. The two lines are: x + y = 3 x + y = 4 They are parallel. (Recall that the test of parallel lines is a1/a2 = b1/b2) They intersect the x axis at x = 3 and x = 4 and at y axis at y = 3 and y = 4. Now there are many ways of getting the distance between them. The first that comes to my mind is using the little triangle abc. ab is dropped perpendicular to the line. Co-ordinates of a are (3, 0) and of c are (4, 0). Ques3.jpg In the little triangle, ac = 1 unit, angle bca is 45 degrees so it is a 45-45-90 triangle. Since ac = 1 unit, ab = 1/\sqrt{2} Recall that the sides in a 45-45-90 are 1:1: \sqrt{2} The distance between the 2 lines = ab = 1/\sqrt{2} or \sqrt{2}/2

DeeptiM · Answer

Thanks Karishma but how did we know angle bca is 45 degrees..
I understood your point of 45-45-90 triangle but how did we arrive at sqrt2/2
Pls help..

mrblack · Answer

Since you know that triangle BCA is a right angle isosceles triangle, you know that bc=ba. Use the 
Pythagorean Theorem to solve for bc.

KarishmaB · Answer

When you draw the line, you notice that its x and y intercept is the same. 
i.e. x + y = 3 intersects x axis at 3 and y axis also at 3. So it forms an isosceles triangle.
Similarly, x + y = 4 intersects x axis at 4 and y axis also at 4. 
It also forms an isosceles triangle so angle bca is 45 degrees.
Since angle abc is 90 (we dropped a perpendicular), angle bac will also be 45 degrees (to make the sum 180). So ab = ac. 
In an isosceles right triangle, the ratio of the sides is  1:1:\sqrt{2}  where  \sqrt{2}  is the hypotenuse.
Since we know that the hypotenuse is actually 1, the measure of equal sides will be 1/ \sqrt{2} . Multiply and divide this by  \sqrt{2}  to get  \sqrt{2} /2.

fluke · Answer

Distance between two parallel lines y=mx+b and y=mx+c can be found by the formula: D=\frac{|b-c|}{\sqrt{m^2+1}} Here: x+y=3 || y=-x+3 --------------------1 2x+2y=8 || y=-x+4 ---------------------2 D=\frac{|b-c|}{\sqrt{m^2+1}} b=3 c=4 m=-1 D=\frac{|3-4|}{\sqrt{(-1)^2+1}} D=\frac{1}{\sqrt{2}} Formula extracted from: math-coordinate-geometry-87652.html

DeeptiM · Answer

Thanks for being patient with me Karishma..+1 for you

sjavvadi · Answer

The two lines are parallel lines. Once we know that, we can use the distance between parallel lines formula. Or if you prefer drawing figures, easier to draw a line from origin perpendicular to both these lines and calculate the distance.

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Distance b/w 2 lines

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Distance b/w 2 lines

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