Question

In the figure above, $3<a<5$ and $6<b<8$. Which of the following represents all possible values of c?

• A. $0<c<2$
• B. $1<c<3$
• C. $0<c<13$
• D. $1<c<13$
• E. $3<c<13$

Answer 1

Answer: (D)

$1<c<13$

You need to know the third-side rule for triangles to solve this question:

The length of the third side of a triangle is less than the sum of the lengths of the other two sides and greater than the positive difference of the lengths of the other two sides.

Applied to this question, the first part of the rule says that the value of $c$ must be less than $a+b$.

Since we are interested in all possible values of $c$, we need to know the greatest possible value of $a+b$.

With $a<5$ and $b<8,a+b<13$ so that c must be less than $13$.

For the second part of the rule, $c$ must be greater than $ba$. (Note that b is always bigger than a, so that $ba$ is positive.)

We are interested in all possible values of $c$, so we need to know the least possible value of $ba$.

The least value occurs when $b$ is as small as possible and a is as large as possible: $ba>65=1$.

Then, $c$ must be greater than $1$. Putting this together, $1<c<13$, making answer D the correct one.