If RS = 6x, ST = 6, and RT = 9x, what is RT?

Step-by-step solution

1. 1

   Write down what betweenness gives you. $S$ lies between $R$ and $T$, so the parts sum to the whole:

   $RS+ST=RT\;\Longrightarrow\;6x+6=9x$

2. 2

   Solve the linear equation. Move the $6x$ across:

   $6=9x-6x=3x\;\Longrightarrow\;x=2$

3. 3

   Evaluate $RT$, which is what the question requests.

   $RT=9x=9(2)=18$

   Answering $x=2$ instead of $RT=18$ is the standard trap in this problem type.

4. 4

   Check the decomposition. $RS=6(2)=12$ and $ST=6$, so

   $12+6=18=RT\;\checkmark$

   Each length is positive and the whole exceeds each part, so the configuration is valid. Notice the coefficients alone predicted the split: $RS:RT=6:9=2:3$, and indeed $12$ is two-thirds of $18$.