2008 UNCO Math Contest II Problems/Problem 4

Problem

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 $BC$ ?

Solution

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

2008 UNCO Math Contest II (Problems • Answer Key • Resources)
Preceded by Problem 3	Followed by Problem 5
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2008 UNCO Math Contest II Problems/Problem 4

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