@TSS-Graviser Hi there, the \(13a\) and \(12a\) come from the fact that we know the ratios of the \( \textcolor{green}{\text{green}}\) and \(\textcolor{blue}{\text{blue}}\) sides. How do we know the ratio of the \( \textcolor{green}{\text{green}}\) and \(\textcolor{blue}{\text{blue}}\) sides? From the information given in the question, that the yellow segment is 52% of the vertical side.

temp-m2d13-y-2-screenshot.png

Since \(52%\) equals the fraction \(\frac{52}{100},\) which equals \(\frac{4 \times 13}{4 \times 25},\) which reduces to \(\frac{13}{25},\) this means that the other piece of the vertical line under the yellow arrow is \(1 - \frac{13}{25} = \frac{12}{25} \) of the whole vertical side.

That's how we got this picture here:

M2W4D13-y-part-2-angle-bisector-solution1-50-percent.png

Do you see that Angle Bisector Theorem here tells us that the ratio of the \( \textcolor{green}{\text{green}}\) and \(\textcolor{blue}{\text{blue}}\) sides is \( 13 : 12\) ?

We can't say for sure that the \( \textcolor{green}{\text{green}}\) side is \(13\) long and the \(\textcolor{blue}{\text{blue}}\) side is \(12\) long, but we just know that the length of the \( \textcolor{green}{\text{green}}\) side divided by the length of the \(\textcolor{blue}{\text{blue}}\) side is equal to \(\frac{13}{12},\) so we say that the \( \textcolor{green}{\text{green}}\) side has length \(13\) units of some amount, which is \(13a,\) and the \(\textcolor{blue}{\text{blue}}\) side has length \(12\) units of some amount, which is \(12a.\)