Question - Finding Length of Segment in Triangle using Proportionality Theorem
Solution:
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.
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