Question

For triangle ABC, AB = 3 cm and BC = 5 cm.Which could be the measure of AC?A 2 cmB 4 cmC 8 cmD 15 cm

Answer 1

2, 4 and 8

Step-by-step explanation

We have that in a triangle ABC, AB = 3 cm and BC = 5 cm.

To find the possible length of AC, we can apply the triangle inequality theorem.

It states that in any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.

This means that:

`\begin{gathered} AB\text{ + AC }\ge\text{ BC} \\ \text{and } \\ AB\text{ + BC }\ge\text{ AC} \\ \text{and} \\ AC\text{ + BC }\ge\text{ AB} \end{gathered}`

So, we have that:

`\begin{gathered} 3\text{ + AC }\ge\text{ 5 }\Rightarrow\text{ AC }\ge\text{ 2} \\ 3\text{ + 5 }\ge\text{ AC }\Rightarrow\text{ AC }\leq\text{ 8} \\ AC\text{ + 5 }\ge3\Rightarrow\text{ AC }\ge\text{ -2} \end{gathered}`

We have to disregard the third line, since the length of a triangle side can only be positive.

So, using the first 2 lines, we see that:

`2\text{ }\leq\text{ AC }\leq\text{ 8}`

This means that from the options, the measure of AC can either be 2, 4 or 8.