I THOUGHT PO SHEN LOH SAID "ANY OLD 30-60-90" RIGHT TRIANGLE. SO WHY IS THE ANSWER TO TO DAY 13 YOUR TURN PART 2 A TRIANGLE THAT ISN'T A 30 60 90. SOMEHOW IT'S A 5 12 13. WOW MAGIC! HOPE YOU HEAR THE SARCASM!!!

]]>

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:

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 \(12a.\)

*When you have a line bisecting an angle in a triangle, that line will cut the opposite side in two parts, the ratios of which are in the same ratio as the other two sides of the triangle!*

So the ratio of the green and purple sides, \(13:12,\) is the same as the ratio of the segments cut from the third side, \(13:12.\)

In fact, if we had used another \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle, then it might not have been obvious that this fact works for any triangle! It might have appeared to be a property of only \(30^{\circ}-60^{\circ}-90^{\circ}\) right triangles.

]]>