# Possible distance b/w points

I am stumped on the following question (at least a part of the question)

The distance from town A to town B is five miles . Town C is six miles from B .Which of the following could be a distance from A to C ? A)11 b)7 c)1

The answer is all of them. I could only figure out 11. How did they get 7 and 1 ?

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You know two things: the line connecting $A$ and $B$ is five miles long, and the line between $B$ and $C$ is six miles long.

You do not know where $C$ is relative to $B$. That means that $C$ must lie on a circle with a radius of 6 miles from $B$.

If $A$ lies directly between $B$ and $C$, then what is the distance from $A$ to $C$?

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Getting 1 is easy:

Say B is 5 miles directly east of A. Also say that C is 6 miles directly west of B. This makes C 1 mile directly west of A.

Getting 7 is a bit trickier and requires some thought:

We know that A is 5 miles away from B and that B is 6 miles away from C. If we were to make a right triangle with 5 on the bottom and 6 on the side, we would get a hypotenuse length of sqrt(61), which is greater than 7. Therefore, we know that the angle of ABC is less than 90 degrees. We also know that there exists a triangle with sides 5, 6, and 7, and so we have our answer.

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the triangle inequality states that $AB\leq BC+AC$ ,$BC\leq AB+AC$ and $AC\leq BC+AB$ If AC=7. If $AC=11$ then $AB+BC=AC$ which means C is in the road between A and B. if $AC=1, then AB+AC=BC which would mean c is in the road between A and B. The problem is that two of the answers make all towns be colinear while the other one makes a proper triangle with sides 5,6,7. - Draw a picture. Say$A$and$B$live on the$x$-axis, with$B$to the right of$A$. You noticed that if$C$also lives on the$x$-axis,$6$miles to the right of$B$, then$C$will be$11$miles from$A$. If$C$lives on the$x$-axis,$6$miles to the left of$B$, then$C$will be$1$mile from$A$. As for$7$, there certainly is a triangle$ABC$with$AB=5$,$BC=6$, and$CA=7$. In general, if we are given three positive real numbers$a$,$b$, and$c$, and the sum of any two of$a$,$b$, and$c$is greater than the third, then there is a triangle with sides$a$,$b$, and$c$. To think about it another way, draw a circle with centre$B$and radius$6$. Draw a circle with centre$A$and radius$7$. These two circles meet (in fact in two places). So there are two points$C$which are distance$6$from$B$and distance$7$from$A\$.

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