When two triangles are similar the corresponding sides are always in the same ratio.
If triangle MNP is similar to traingle XYZ, the lengths of MNP needs to be in the same ratio with the corresponding sides of triangle XYZ
[tex]\frac{MN}{XY}=\frac{NP}{YZ}=\frac{MP}{XZ}[/tex][tex]\frac{MN}{6}=\frac{MP}{7}=\frac{NP}{9}[/tex]
Prove each of the given options lengths to find the one that makes the equation above be true:
Pair corresponding sides ordering its lengths from least XY to greatest YZ
First option:
[tex]\begin{gathered} \frac{12}{6}=\frac{14}{7}=\frac{20}{9} \\ 2=2=2.22 \end{gathered}[/tex]
As the ratios are not the same that cannot be the lengths of MNP
Second option:
[tex]\begin{gathered} \frac{30}{6}=\frac{35}{7}=\frac{45}{9} \\ 5=5=5 \\ \end{gathered}[/tex]
As the ratios are the same that can be the lengths of MNP
Third option:
[tex]\begin{gathered} \frac{13}{6}=\frac{14}{7}=\frac{16}{9} \\ 2.16=2=1.77 \end{gathered}[/tex]
As the ratios are not the same that cannot be the lengths of MNP
Fourth option:
[tex]\begin{gathered} \frac{3}{6}=\frac{4}{7}=\frac{5}{9} \\ 0.5=0.57=0.55 \end{gathered}[/tex]
As the ratios are not the same that cannot be the lengths of MNP
Then, the lengths of MNP could be: 30cm,35cm,45cm
