Question

use the converse of the side - splitter theorem to determine if tu || rs. which statement is true?
line segment tu is parallel to line segment rs because \\( \frac { 32 } { 36 } = \frac { 40 } { 45 } \\)
line segment tu is not parallel to line segment rs because \\( \frac { 32 } { 36 } \
eq \frac { 40 } { 45 } \\)
line segment tu is parallel to line segment rs because \\( \frac { 32 } { 45 } = \frac { 40 } { 36 } \\)
line segment tu is not parallel to line segment rs because \\( \frac { 32 } { 45 } \
eq \frac { 40 } { 36 } \\)

Explanation:

Step1: Recall the converse of the side - splitter theorem

The converse of the side - splitter theorem states that if a line divides two sides of a triangle proportionally, then the line is parallel to the third side. For \(\triangle QRS\) with line \(TU\), we need to check if \(\frac{QT}{TR}=\frac{QU}{US}\)

Step2: Calculate the ratios

Given \(QT = 32\), \(TR=36\), \(QU = 40\), \(US = 45\)
The ratio \(\frac{QT}{TR}=\frac{32}{36}=\frac{8}{9}\) (simplified by dividing numerator and denominator by 4)
The ratio \(\frac{QU}{US}=\frac{40}{45}=\frac{8}{9}\) (simplified by dividing numerator and denominator by 5)

Answer:

Line segment \(TU\) is parallel to line segment \(RS\) because \(\frac{32}{36}=\frac{40}{45}\)