â–³ ABC ~ â–³ RST AD is a median of â–³ ABC RU is a median of â–³ RST
1.
Given
2.
DC = BD and UT=SU
2.
Definition of median
3.
AB/RS = BC/ST
3.
Definition of similar triangles
4.
BC = BD+DC and ST = SU+UT
4.
Segment Addition Postulate
5.
AB/RS = BD+DC/SU+UT
5.
Substitution
6.
AB/RS = BD+BD/SU+SU or 2BD/2SU
6.
Substitution
7.
AB/RS = BD/SU
7.
Simplifying
8.
∠B ≅ ∠S
8.
Definition of similar triangles
9.
â–³ ABD ~ â–³ RSU
9.
SAS Similarity Theorem
10.
AD/RU = AB/RS
10.
Definition of similar triangles
Practice makes perfect
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 â–³ ABC and â–³ RST such that they are similar.
By the definition of similar triangles, we can write the following proportion and angle congruence.
AB/RS = BC/ST and ∠B ≅ ∠S
Next, we draw the medians from vertices A and R.
By the definition of medians, we have the following.
BD&=DC
SU &=UT
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.
AB/RS = BD/SU and ∠B ≅ ∠S
Applying the Side-Angle-Side (SAS) Similarity Theorem we get that â–³ ABD ~ â–³ RSU. Once again, in similar triangles corresponding lengths are proportional. This gives is the following relationship.
AD/RU = AB/RS
Two-Column Proof
Let's summarize the proof we wrote in the following two-column table.
Given: & â–³ ABC ~ â–³ RST
& AD is a median ofâ–³ ABC
& RU is a median ofâ–³ RST
Prove: & ADRU = ABRS
Statements
Reasons
1.
â–³ ABC ~ â–³ RST AD is a median of â–³ ABC RU is a median of â–³ RST
