Exercise 19 Page 506

We are asked to write a two-column proof of the following theorem.

Theorem $7.10$

If two triangles are similar, the lengths of corresponding medians are proportional to the lengths of corresponding sides.

Let's consider $△ABC$ and $△RST$ such that they are similar.

By the definition of similar triangles, we can write the following proportion and angle congruence. Next, we draw the medians from vertices $A$ and $R.$ By the definition of medians, we have the following. Also, the Segment Addition Postulate tells us that $BC$ can be expressed as a sum $BD+DC.$ Similarly, $ST=SU+UT.$ Let's substitute these equalities into the proportion we wrote earlier. As a result, we have the following proportion and angle congruence. Applying the Side-Angle-Side (SAS) Similarity Theorem we get that $△ABD\sim △RSU.$ Once again, in similar triangles corresponding lengths are proportional. This gives is the following relationship.

Two-Column Proof

Let's summarize the proof we wrote in the following two-column table.

Statements	Reasons
$△ABC\sim △RST$ $\overline{AD}$ is a median of $△ABC$ $\overline{RU}$ is a median of $△RST$	Given
$DC=BD$ and $UT=SU$	Definition of median
$\frac{AB}{RS}=\frac{BC}{ST}$	Definition of similar triangles
$BC=BD+DC$ and $ST=SU+UT$	Segment Addition Postulate
$\frac{AB}{RS}=\frac{BD+DC}{SU+UT}$	Substitution
$\frac{AB}{RS}=\frac{BD+BD}{SU+SU}$ or $\frac{2BD}{2SU}$	Substitution
$\frac{AB}{RS}=\frac{BD}{SU}$	Simplifying
$\angle B\cong \angle S$	Definition of similar triangles
$△ABD\sim △RSU$	SAS Similarity Theorem
$\frac{AD}{RU}=\frac{AB}{RS}$	Definition of similar triangles