Chapter 7.6, Problem 18WE

To determine

To Prove: The corollary of the triangle proportionality theorem.

Explanation of Solution

Proportional lengths: Point L and M lie on AB and CD, respectively. If $\frac{AL}{LB}=\frac{CM}{MD}$ , then AB and CD are divided proportionally.

Corollary: If three lines intersect two transversals, then they divide the traversals proportionally.

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

Then, $\frac{RS}{ST}=\frac{XY}{YZ}$

Proof: Draw a line TX, such that it intersects SY at N. Note that SY is parallel to one side of the triangle RTX, and also to one side of the triangle TXZ.

Now, use triangle proportionality theorem to both of these triangles,

Triangle Proportionality Theorem:

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

If $\text{Δ}RTX:\overrightarrow{SN}\parallel \overrightarrow{RX}$

Then, $\frac{RS}{ST}=\frac{XN}{NT}\dots \dots (1)$

And, if $\text{Δ}TXZ:\overline{NY}\parallel \overline{TZ}$

Then $\frac{XN}{NT}=\frac{XY}{YZ}\dots \dots$ (2)

Use both the equations (1) and (2),

  $\frac{RS}{ST}=\frac{XY}{YZ}$

Hence, if three lines intersect two transversals, then they divide the transversals proportionally.

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

Then, $\frac{RS}{ST}=\frac{XY}{YZ}$