Difference between revisions of "2009 AMC 10B Problems/Problem 12"

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Each of the triangles that we can make has exactly one side lying on one of the two parallel lines. If we pick this side to be the base, the height will always be the same - it will be the distance between the two lines.
 
Each of the triangles that we can make has exactly one side lying on one of the two parallel lines. If we pick this side to be the base, the height will always be the same - it will be the distance between the two lines.
  
Hence each area is uniquely determined by the length of the base. And it can easily be seen, that the only possible base lengths are <math>1</math>, <math>2</math>, and <math>3</math>. Therefore there are only <math>\boxed{3}</math> possible values for the area.
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Hence each area is uniquely determined by the length of the base. And it can easily be seen, that the only possible base lengths are <math>1</math>, <math>2</math>, and <math>3</math>. Therefore there are only <math>\boxed{(A)3}</math> possible values for the area.
  
 
(To be more precise in the last step, the possible base lengths are <math>AB=BC=CD=EF=1</math>, <math>AC=BD=2</math>, and <math>AD=3</math>.)
 
(To be more precise in the last step, the possible base lengths are <math>AB=BC=CD=EF=1</math>, <math>AC=BD=2</math>, and <math>AD=3</math>.)

Latest revision as of 00:50, 16 January 2020

Problem

Distinct points $A$, $B$, $C$, and $D$ lie on a line, with $AB=BC=CD=1$. Points $E$ and $F$ lie on a second line, parallel to the first, with $EF=1$. A triangle with positive area has three of the six points as its vertices. How many possible values are there for the area of the triangle?

$\text{(A) } 3 \qquad \text{(B) } 4 \qquad \text{(C) } 5 \qquad \text{(D) } 6 \qquad \text{(E) } 7$

Solution

Consider the classical formula for triangle area: $\frac 12 \cdot b \cdot h$. Each of the triangles that we can make has exactly one side lying on one of the two parallel lines. If we pick this side to be the base, the height will always be the same - it will be the distance between the two lines.

Hence each area is uniquely determined by the length of the base. And it can easily be seen, that the only possible base lengths are $1$, $2$, and $3$. Therefore there are only $\boxed{(A)3}$ possible values for the area.

(To be more precise in the last step, the possible base lengths are $AB=BC=CD=EF=1$, $AC=BD=2$, and $AD=3$.)

See Also

2009 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 11
Followed by
Problem 13
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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