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Difference between revisions of "2008 AMC 10B Problems/Problem 7"
Line 1: Line 1:
 
==Problem==
 
==Problem==
−
An equilateral triangle of side length 10 is completely filled with in by non overlapping equilateral triangles of side length 1. How many small triangles are required?  
+
An equilateral triangle of side length <math>10</math> is completely filled in by non-overlapping equilateral triangles of side length <math>1</math>. How many small triangles are required?
  
−
A) 10   B) 25   C) 100   D) 250   E) 1000
+
<math>\mathrm{(A)}\ 10\qquad\mathrm{(B)}\ 25\qquad\mathrm{(C)}\ 100\qquad\mathrm{(D)}\ 250\qquad\mathrm{(E)}\ 1000</math>
  
 
==Solution==
 
==Solution==
−
{{solution}}
+
 
 +
<asy>
 +
unitsize(0.5cm);
 +
defaultpen(0.8);
 +
for (int i=0; i<10; ++i) { draw( (i*dir(60)) -- ( (10,0) + (i*dir(120)) ) ); }
 +
for (int i=0; i<10; ++i) { draw( (i*dir(0)) -- ( 10*dir(60) + (i*dir(-60)) ) ); }
 +
for (int i=0; i<10; ++i) { draw( ((10-i)*dir(60)) -- ((10-i)*dir(0)) ); }
 +
</asy>
 +
 
 +
The number of triangles is <math>1+3+\dots+19 = \boxed{100}</math>.
  
 
==See also==
 
==See also==
 
{{AMC10 box|year=2008|ab=B|num-b=6|num-a=8}}
 
{{AMC10 box|year=2008|ab=B|num-b=6|num-a=8}}

Revision as of 16:25, 25 January 2009

Problem

An equilateral triangle of side length $10$ is completely filled in by non-overlapping equilateral triangles of side length $1$. How many small triangles are required?

$\mathrm{(A)}\ 10\qquad\mathrm{(B)}\ 25\qquad\mathrm{(C)}\ 100\qquad\mathrm{(D)}\ 250\qquad\mathrm{(E)}\ 1000$

Solution

[asy] unitsize(0.5cm); defaultpen(0.8); for (int i=0; i<10; ++i) { draw( (i*dir(60)) -- ( (10,0) + (i*dir(120)) ) ); } for (int i=0; i<10; ++i) { draw( (i*dir(0)) -- ( 10*dir(60) + (i*dir(-60)) ) ); } for (int i=0; i<10; ++i) { draw( ((10-i)*dir(60)) -- ((10-i)*dir(0)) ); } [/asy]

The number of triangles is $1+3+\dots+19 = \boxed{100}$.

See also

2008 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 6 		Followed by
Problem 8
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions
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