Question

if ( uv = y ) and ( rt = y - 41 ), what is ( rt )?

Explanation:

Step1: Use the mid - segment theorem

In a triangle, the mid - segment (a segment connecting the midpoints of two sides of a triangle) is half the length of the third side. Here, \(RT\) is a mid - segment of \(\triangle SUV\) (since \(R\) and \(T\) are midpoints), so \(RT=\frac{1}{2}UV\).
Given \(UV = y\) and \(RT=y - 41\), we substitute into the mid - segment formula: \(y-41=\frac{1}{2}y\).

Step2: Solve the equation for \(y\)

Subtract \(\frac{1}{2}y\) from both sides of the equation \(y-41=\frac{1}{2}y\):
\(y-\frac{1}{2}y-41=\frac{1}{2}y-\frac{1}{2}y\)
\(\frac{2y - y}{2}-41 = 0\)
\(\frac{y}{2}-41=0\)
Add \(41\) to both sides: \(\frac{y}{2}=41\).
Multiply both sides by \(2\) to get \(y = 82\).

Step3: Find the value of \(RT\)

Since \(RT=y - 41\) and \(y = 82\), substitute \(y\) into the formula for \(RT\):
\(RT=82-41\).

Answer:

\(41\)