I've been stumped on this for like an hour!! any help greatly appreciated:
In triangle ABC, AB=20 cm, BC=7 cm, and CA=15 cm. When side BC is extended to point D, triangle DAB is similar to triangle DCA. What is the integer value of side DC?
I've been stumped on this for like an hour!! any help greatly appreciated:
In triangle ABC, AB=20 cm, BC=7 cm, and CA=15 cm. When side BC is extended to point D, triangle DAB is similar to triangle DCA. What is the integer value of side DC?
Hi ... If you draw a diagram, you'll see that triangle DAB and DCA are sort of "anti-similar", that is, when you list corresponding vertices, you'll be reading the vertices of one triangle clockwise and the other counter-clockwise. Your book or teacher has listed the vertices correctly ... D to D, A to C, B to A. you could probably get away without explaining or even understanding why this must be so, but if you can understand why, that will be helpful for you.
Now all you have to do is write out the proportions, remembering that corresponding sides of the similar triangles are in proportion. Plug in the values you know. Remember that, for example, AD=DA, CD=DC etc, doesn't matter the order (we're using absolute values here, not directed distances).
See the pdf.
I managed to make another pdf from GSP, showing the situation. When you construct a cevian so that there are two similar triangles, the similarity will always be opposite in the sense I explained. There's a word for it, though I can't remember it right now.
What I did was essentially the same as what our friend idbutt did. The only difference was that I did another substitution ... I thought it would prevent any confusion about whether, for example, DC = CD (they do).
I'm including another pdf where I use the same variables that he did. Maybe this will be clearer. The distance you want is DC = 9. That's the amount you have to extend BC past C to get to D.
understanding why this must be so, but if you can understand why
, that will be helpful for you.