2. TU, UV and TV are mid-segments
3. Definition of mid-segment
4. Division property of equality
6. SSS similarity theorem
Solution:
Step 1: Given
T is the midpoint of QR.
U is the midpoint of QS.
V is the midpoint of RS.
Step 2: Mid-segments connects midpoints of opposite sides.
TU, UV and TV are mid-segments.
Step 3: By definition of mid-segment
A triangle mid-segment is parallel to the third side of the triangle and is half of the length of the third side.
[tex]T U=\frac{1}{2} R S,[/tex] [tex]{U V}=\frac{1}{2} {Q} R[/tex] and [tex]V T=\frac{1}{2} S Q[/tex]
Step 4: By division property of equality
Divide first segment by RS, second segment by QR and third segment by SQ.
[tex]$\frac{T U}{R S}=\frac{1}{2}, \ \frac{U V}{Q R}=\frac{1}{2}, \ \frac{V T}{S Q}=\frac{1}{2}[/tex]
Step 5: By transitive property
[tex]$\frac{T U}{R S}=\frac{U V}{Q R}=\frac{V T}{S Q}[/tex]
Step 6: From the above steps, the sides of the triangle are congruent.
By SSS similarity theorem,
[tex]\Delta \text { QRS } \sim \Delta \text { VUT }[/tex]
Hence proved.
