Chapter 7.6, Problem 27WE

To determine

To prove: The Triangle Proportionality Theoremfor the given figure.

Explanation of Solution

Given information:

  

According to the Triangle proportionality theorem:

If the two transverals are intersected by three parallel lines, then the parallel lines divide the transversals proportionally.

Further, consider the diagram as follows:

  

It shown that:

  $\overrightarrow{RX}\mid \overrightarrow{SY}\parallel \overrightarrow{TZ}$

Draw the line TX. This intersects the line SY at O.

Therefore, look at the diagram:

  

Consider the following theorem:

The sides of triangle are divided proportionally ,if a parallel line to one side of triangle intersects other side.

Therefore, in the triangle $\text{Δ}RXT$ ,

  $\frac{RS}{ST}=\frac{XO}{OT}$

Also, in the triangle $\text{Δ}XZT$ ,

  $\frac{XY}{YZ}=\frac{XO}{OT}$

Compare and get the expression as follows:

  $\frac{RS}{ST}=\frac{XY}{YZ}$