Difference between revisions of "2000 AMC 10 Problems/Problem 5"
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(a) Clearly <math>AB</math> does not change, and <math>MN=\frac{1}{2}AB</math>, so <math>MN</math> doesn't change either. | (a) Clearly <math>AB</math> does not change, and <math>MN=\frac{1}{2}AB</math>, so <math>MN</math> doesn't change either. | ||
| − | (b) Obviously, the perimeter changes. For example, imagine if | + | (b) Obviously, the perimeter changes. For example, imagine if P was extremely far to the left. |
(c) The area clearly doesn't change, as both the base <math>AB</math> and its corresponding height remain the same. | (c) The area clearly doesn't change, as both the base <math>AB</math> and its corresponding height remain the same. | ||
Revision as of 17:01, 29 October 2017
Problem
Points 
and 
are the midpoints of sides 
and 
of 
. As 
moves along a line that is parallel to side 
, how many of the four quantities listed below change?
(a) the length of the segment 
(b) the perimeter of
(c) the area of
(d) the area of trapezoid 
![[asy] draw((2,0)--(8,0)--(6,4)--cycle); draw((4,2)--(7,2)); draw((1,4)--(9,4),Arrows); label("$A$",(2,0),SW); label("$B$",(8,0),SE); label("$M$",(4,2),W); label("$N$",(7,2),E); label("$P$",(6,4),N); [/asy]](http://latex.artofproblemsolving.com/0/b/6/0b66755a2c42a83a93ad849254ee5df3266bb45c.png)
Solution
(a) Clearly does not change, and 
, so doesn't change either.
(b) Obviously, the perimeter changes. For example, imagine if P was extremely far to the left.
(c) The area clearly doesn't change, as both the base and its corresponding height remain the same.
(d) The bases and do not change, and neither does the height, so the area of the trapezoid remains the same.
Only 
quantity changes, so the correct answer is 
.
See Also
| 2000 AMC 10 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 4 |
Followed by Problem 6 | |
| 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. 
