Finding Length of Segment in Triangle using Proportionality Theorem
The image shows a large triangle \( \triangle RQU \) with a line segment \( \overline{ST} \) cutting it proportionally, which means \( \overline{ST} \) is parallel to \( \overline{QU} \). When a line segment cuts two sides of a triangle proportionally, the Triangle Proportionality Theorem (also known as the Basic Proportionality Theorem or Thales' Theorem) states that it divides those sides proportionally.
Given in the diagram:
- The length of \( \overline{RQ} \) is 16.
- The length of \( \overline{QS} \) is 5.
- The length of \( \overline{TU} \) is 8.
We need to find the length of \( \overline{RS} \).
Let's denote the length of \( \overline{RS} \) as \( x \). So \( \overline{RT} = x + 5 \).
According to the Triangle Proportionality Theorem:
\[
\frac{RQ}{QS} = \frac{RT}{TU}
\]
Now we can plug the values into the equation:
\[
\frac{16}{5} = \frac{x + 5}{8}
\]
Now, solve for \( x \):
\[
128 = 5x + 25
\]
Subtract 25 from both sides:
\[
128 - 25 = 5x
\]
\[
103 = 5x
\]
Now divide both sides by 5 to solve for \( x \):
\[
x = \frac{103}{5}
\]
\[
x = 20.6
\]
So, the length of \( \overline{RS} \) is 20.6.
