Okay this question is incredibly confusing for me. So i know triangle CHR and triangle DOR are similar because they share two angles.
Here is an explanation I found to this question:
It may not seem obvious at first that the two triangles are similar because DO is not parallel to CH. However, you are given one pair of angles congruent, and both triangles contain angle R. Therefore, they are similar.
The proper proportion to set up is: $\frac{RD}{HR} = \frac{RO}{CR}$. And $HR = 4 + 6 = 10$.
So $\frac{4}{6} = \frac{10}{CR}$
$4*CR = 60$
$CR = 15$
$CD = CR – DR = 15 – 4 = 11$.
Don't forget to subtract the length of RD from the side of the triangle because they are looking for $CD$.
Now here is my problem. I rotated the triangles different ways so that I can find the corresponding sides and I keep finding that OD corresponds with HR unlike the explanation who said RD corresponds with HR.
I wasnt convinced at first but the person got the correct answer however I still don't see it. Any ideas or suggestions on how to know which sides correspond to the other sides?
Best Answer
Note there is an inconsistency with the proportions the answer sets up. They say $\frac{RD}{HR}=\frac{RO}{CR}$, but then they plug in the numbers according to the proportion $\frac{RD}{RO}=\frac{HR}{CR}$. Both proportions are correct, so the answer $CR=15$ is correct. Correct, but confusing. So if $CR=15$, the correct answer is 11.
Now regarding which side corresponds with which side. Instead of looking at the sides, I would look at the angles. In $\triangle CHR$, the only obtuse angle is the angle at vertex $H$. Similarly, in $\triangle ODR$, the only obtuse angle is at vertex $D$. So one pair of corresponding sides would be from the shared angle $R$ to the obtuse angle; therefore $RH$ corresponds with $RD$.
Hope that helps.