A way to prove similar triangles is by SSS, side-side-side. If the measures of corresponding sides are known, then their proportionality can be calculated.
If two triangles have three pairs of sides in the same ratio, then the triangles are similar.
Let us name the sides of the triangle using their lengths. We have:
[tex]\begin{gathered} s=\text{ smallest side} \\ m=\text{ middle side} \\ l=\text{ longest side} \end{gathered}[/tex]
Therefore, for triangle 1, we have:
[tex]\begin{gathered} s_1=7 \\ m_1=24 \\ l_1=25 \end{gathered}[/tex]
and for triangle 2. we have:
[tex]\begin{gathered} s_2=5 \\ m_2=12 \\ l_2=13 \end{gathered}[/tex]
Following the SSS rule, we should have:
[tex]s_1\colon s_2=m_1\colon m_2=l_1\colon l_2[/tex]
First ratio:
[tex]s_1\colon s_2=7\colon5[/tex]
Second ratio:
[tex]m_1\colon m_2=24\colon12=2\colon1[/tex]
Third ratio:
[tex]l_1\colon l_2=25\colon13[/tex]
CONCLUSION:
We can see that all the ratios are not equal.
Therefore, the triangles are NOT SIMILAR.
