Sure, I can guide you through the solution in a detailed, step-by-step manner:
1. Understand the problem:
- We are given three line segments: [tex]\( RS \)[/tex], [tex]\( ST \)[/tex], and [tex]\( RT \)[/tex].
- The point [tex]\( S \)[/tex] is between points [tex]\( R \)[/tex] and [tex]\( T \)[/tex].
- The lengths of these segments are given in terms of [tex]\( x \)[/tex]:
[tex]\[
RS = 2x + 10, \quad ST = x - 4, \quad RT = 21
\][/tex]
2. Form an equation based on the given information:
- Since point [tex]\( S \)[/tex] is between [tex]\( R \)[/tex] and [tex]\( T \)[/tex], the length [tex]\( RT \)[/tex] is the sum of lengths [tex]\( RS \)[/tex] and [tex]\( ST \)[/tex]. Hence, we can write:
[tex]\[
RS + ST = RT
\][/tex]
- Substituting the given expressions for [tex]\( RS \)[/tex], [tex]\( ST \)[/tex], and [tex]\( RT \)[/tex], we get:
[tex]\[
(2x + 10) + (x - 4) = 21
\][/tex]
3. Simplify the equation:
- Combine like terms:
[tex]\[
2x + x + 10 - 4 = 21
\][/tex]
- Simplify further:
[tex]\[
3x + 6 = 21
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
- Subtract 6 from both sides:
[tex]\[
3x = 15
\][/tex]
- Divide both sides by 3:
[tex]\[
x = 5
\][/tex]
5. Find the lengths of [tex]\( RS \)[/tex], [tex]\( ST \)[/tex], and [tex]\( RT \)[/tex]:
- Substitute [tex]\( x = 5 \)[/tex] into the expression for [tex]\( RS \)[/tex]:
[tex]\[
RS = 2x + 10 = 2(5) + 10 = 10 + 10 = 20
\][/tex]
- Substitute [tex]\( x = 5 \)[/tex] into the expression for [tex]\( ST \)[/tex]:
[tex]\[
ST = x - 4 = 5 - 4 = 1
\][/tex]
- Since [tex]\( RT = 21 \)[/tex] is already given and is the sum of [tex]\( RS \)[/tex] and [tex]\( ST \)[/tex], verify:
[tex]\[
RT = RS + ST = 20 + 1 = 21
\][/tex]
6. Summarize the results:
- The solution for [tex]\( x \)[/tex] is [tex]\( x = 5 \)[/tex].
- The lengths of the segments are:
[tex]\[
RS = 20, \quad ST = 1, \quad RT = 21
\][/tex]
So, the equation formed is [tex]\( 3x + 6 = 21 \)[/tex], solving this we find [tex]\( x = 5 \)[/tex]. Substituting [tex]\( x \)[/tex] back, we get [tex]\( RS = 20 \)[/tex], [tex]\( ST = 1 \)[/tex], and the total length [tex]\( RT = 21 \)[/tex] matches the given value.
