Difference between revisions of "2000 AMC 10 Problems/Problem 5"
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(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>. | (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) | + | (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. | (c) The area clearly doesn't change, as both the base <math>AB</math> and its corresponding height remain the same. |
Revision as of 01:24, 27 July 2023
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
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
.
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 |
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