2008 UNCO Math Contest II Problems/Problem 4


In the figure there are $8$ line segments drawn from vertex $A$ to the base $BC$ (not counting the segments $AB$ or $AC$).

[asy] for (int x=0;x<11;++x){ draw((5,15)--(x,0),dot); } draw((0,0)--(10,0),black); draw((10/6,5)--(10-10/6,5),black); draw((20/6,10)--(10-20/6,10),black); MP("A",(5,15),N);MP("B",(0,0),W);MP("C",(10,0),E); [/asy]

(a) Determine the total number of triangles of all sizes.

(b) How many triangles are there if there are $n$ lines drawn from $A$ to $n$ interior points on $BC$?


(a) $3\binom{10}{2}$ (b) $3\binom{n+2}{2}=\frac{3(n+1)(n+2)}{2}$

See Also

2008 UNCO Math Contest II (ProblemsAnswer KeyResources)
Preceded by
Problem 3
Followed by
Problem 5
1 2 3 4 5 6 7 8 9 10
All UNCO Math Contest Problems and Solutions