In the given similar traingles ABC and MNO
From the properties of similar triangle :
The ratio of corresponding sides of similar triangle are always equal
In the triangle ABC and MNO
[tex]\frac{AB}{MN}=\frac{BC}{NO}=\frac{CA}{OM}[/tex]In the given figure, we have : AB = 12, BC = 10, NO = 3
[tex]\begin{gathered} \frac{AB}{MN}=\frac{BC}{NO}=\frac{CA}{OM} \\ \frac{12}{MN}=\frac{10}{3}=\frac{CA}{OM} \\ \text{ simplify the first two and solve for MN} \\ \frac{12}{MN}=\frac{10}{3} \\ MN=\frac{3\times12}{10} \\ MN=\frac{36}{10} \\ MN=3.6\text{ cm} \end{gathered}[/tex]In triangle ABC apply pythagoras for the side AC
[tex]\begin{gathered} \text{ Hypotenuse}^2=Perpendicular^2+Base^2 \\ AC^2=AB^2+BC^2 \\ AC^2=12^2+10^2 \\ AC^2=144\text{ + 100} \\ AC^2=244 \\ AC=\sqrt[]{244} \\ AC=15.62\text{ cm} \end{gathered}[/tex]Now apply the corresponding ratio :
[tex]\begin{gathered} \frac{AB}{MN}=\frac{BC}{NO}=\frac{CA}{OM} \\ \frac{12}{3.6}=\frac{10}{3}=\frac{15.62}{OM} \\ \text{ simplify the last two and solve for OM :} \\ \frac{10}{3}=\frac{15.62}{OM} \\ OM=\frac{15.62\times3}{10} \\ OM=4.686\text{ cm} \end{gathered}[/tex]OM = 4.69 cm
In triangle MNO
MN = 3.6 cm, NO = 3 cm, OM = 4.69 cm
Answer :
MN = 3.6 cm,
NO = 3 cm,
OM = 4.69 cm
