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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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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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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.

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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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