Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

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

m (→‎Video Solution)
m (Fixing my solution)
 
(2 intermediate revisions by 2 users not shown)
Line 16: Line 16:
 
[[Category: Introductory Geometry Problems]]
 
[[Category: Introductory Geometry Problems]]
  
−
== Solution ==
+
== Solution 1 ==
  
 
Consider the classical formula for triangle area: <math>\frac 12 \cdot b \cdot h</math>.  
 
Consider the classical formula for triangle area: <math>\frac 12 \cdot b \cdot h</math>.  
Line 24: Line 24:
  
 
(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>.)
 +
 +
== Solution 2 ==
 +
 +
No matter what how we draw a triangle by selecting three non-linear points, its height will always remain the same. Therefore, we will only get different areas with different base-lengths. The possibilities are <math>1</math>, <math>2</math>, and <math>3</math> units for a total of <math>\boxed{\text{(A) } 3}</math>.
 +
 +
~MathFun1000
  
 
==Video Solution==
 
==Video Solution==
−
https://www.youtube.com/watch?v=dQw4w9WgXcQ
+
https://youtu.be/57B-aRqONv8
  
 
~savannahsolver
 
~savannahsolver

Latest revision as of 11:41, 11 August 2021

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 1

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 $AD=3$.)

Solution 2

No matter what how we draw a triangle by selecting three non-linear points, its height will always remain the same. Therefore, we will only get different areas with different base-lengths. The possibilities are $1$, $2$, and $3$ units for a total of $\boxed{\text{(A) } 3}$.

~MathFun1000

Video Solution

https://youtu.be/57B-aRqONv8

~savannahsolver

See Also

2009 AMC 10B (Problems • Answer Key • Resources)
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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2009_AMC_10B_Problems/Problem_12&oldid=160035"