Chapter 8.5, Problem 25PSB

To determine

To show a line bisects another line.

Explanation of Solution

Given information:

  $\overleftrightarrow{RU}$ bisects $\overline{QS}$ .

  $\overleftrightarrow{RU}$ is parallel to $\overline{ST}$ .

Formula used:

The below properties and theorem are used:

Bisector divides segment into two congruent parts.

Lines parallel to sides of triangle intersects sides and divides sides proportionally.

If a segment divides segment into two congruent parts, it bisects the segment.

Means extremes product theorem:

If a product of a pair of not zero numbers is equal to the product of another pair of not zero numbers, then either pair of numbers be made the extremes, and the other pair the means, of proportion.

Calculation:

  

It is given that,

  $\overleftrightarrow{RU}$ bisects $\overline{QS}$

  $\overleftrightarrow{RU}$ is parallel to $\overline{ST}$

Bisector divides segment into two congruent parts.

  $\overline{QR}\cong \overline{RS}$

Lines parallel to sides of triangle intersects sides and divides sides proportionally.

  $\frac{QR}{RS}=\frac{QU}{UT}$

By substitution property, we get

  $\frac{QR}{QR}=\frac{QU}{UT}$

Applying means extremes product theorem, we get

  $\left(QU\right)\left(QR\right)=\left(UT\right)\left(QR\right)$

By division property, we get

  $\left(QU\right)=\left(UT\right)$

If a segment divides segment into two congruent parts, it bisects the segment.

  $\overleftrightarrow{RU}$ bisects $\overline{QT}$ .