Difference between revisions of "2000 AMC 10 Problems/Problem 5"
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| − | + | ==Problem== | |
| − | + | Points <math>M</math> and <math>N</math> are the midpoints of sides <math>PA</math> and <math>PB</math> of <math>\triangle PAB</math>. As <math>P</math> moves along a line that is parallel to side <math>AB</math>, how many of the four quantities listed below change? | |
| − | ( | + | (a) the length of the segment <math>MN</math> |
| − | ( | + | (b) the perimeter of <math>\triangle PAB</math> |
| − | Only <math>1</math> changes, so <math>\boxed{\text{B}}</math>. | + | (c) the area of <math>\triangle PAB</math> |
| + | |||
| + | (d) the area of trapezoid <math>ABNM</math> | ||
| + | |||
| + | <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> | ||
| + | |||
| + | <math>\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4</math> | ||
| + | |||
| + | ==Solution== | ||
| + | |||
| + | (a) Triangles <math>ABP</math> and <math>MNP</math> are similar, and since <math>PM=\frac{1}{2}AP</math>, <math>MN=\frac{1}{2}AB</math>. | ||
| + | |||
| + | (b) We see 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. | ||
| + | |||
| + | (d) The bases <math>AB</math> and <math>MN</math> do not change, and neither does the height, so the area of the trapezoid remains the same. | ||
| + | |||
| + | Only <math>1</math> quantity changes, so the correct answer is <math>\boxed{\text{B}}</math>. | ||
| + | |||
| + | ==Video Solution by Daily Dose of Math== | ||
| + | |||
| + | https://youtu.be/yf4eWFAmmd0?si=voNN42OoTcS8Z3sz | ||
| + | |||
| + | ~Thesmartgreekmathdude | ||
| + | |||
| + | ==See Also== | ||
| + | |||
| + | {{AMC10 box|year=2000|num-b=4|num-a=6}} | ||
| + | {{MAA Notice}} | ||
| + | [[Category:Introductory Geometry Problems]] | ||
Latest revision as of 23:38, 14 July 2024
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) Triangles 
and 
are similar, and since 
, 
.
(b) We see 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 
.
Video Solution by Daily Dose of Math
https://youtu.be/yf4eWFAmmd0?si=voNN42OoTcS8Z3sz
~Thesmartgreekmathdude
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. 
