Difference between revisions of "2008 UNCO Math Contest II Problems/Problem 4"
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== Problem == | == Problem == | ||
| + | In the figure there are <math>8</math> line segments drawn from | ||
| + | vertex <math>A</math> to the base <math>BC</math> (not counting the segments <math>AB</math> or <math>AC</math>). | ||
| − | + | <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 | + | (a) Determine the total number of triangles of all sizes. |
| − | (b) | + | (b) How many triangles are there if there are <math>n</math> lines |
| + | drawn from <math>A</math> to <math>n</math> interior points on <math>BC</math>? | ||
== Solution == | == Solution == | ||
| − | + | (a) <math>3\binom{10}{2}</math> (b) <math>3\binom{n+2}{2}=\frac{3(n+1)(n+2)}{2}</math> | |
== See Also == | == See Also == | ||
{{UNCO Math Contest box|n=II|year=2008|num-b=3|num-a=5}} | {{UNCO Math Contest box|n=II|year=2008|num-b=3|num-a=5}} | ||
| − | [[Category:Intermediate | + | [[Category:Intermediate Geometry Problems]] |
Latest revision as of 01:01, 13 January 2019
Problem
In the figure there are 
line segments drawn from
vertex 
to the base 
(not counting the segments 
or 
).
![[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]](http://latex.artofproblemsolving.com/0/1/2/012fc4457f36f81cdf88876917f2cd36048e3faa.png)
(a) Determine the total number of triangles of all sizes.
(b) How many triangles are there if there are 
lines
drawn from to interior points on ?
Solution
(a) 
(b) 
See Also
| 2008 UNCO Math Contest II (Problems • Answer Key • Resources) | ||
| 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 | ||
