Difference between revisions of "1995 AJHSME Problems/Problem 9"
Talkinaway (talk | contribs) (Created page with "==Problem== Three congruent circles with centers <math>P</math>, <math>Q</math>, and <math>R</math> are tangent to the sides of rectangle <math>ABCD</math> as shown. The circle...") |
|||
(One intermediate revision by one other user not shown) | |||
Line 29: | Line 29: | ||
Draw a horizontal line <math>PQR</math> that extends to the sides of the rectangle. This line is <math>2</math> diameters long, so it has length <math>4\cdot 2 = 8</math>. It is parallel and congruent to <math>AB</math>, so the width of the rectangle is <math>8</math>. | Draw a horizontal line <math>PQR</math> that extends to the sides of the rectangle. This line is <math>2</math> diameters long, so it has length <math>4\cdot 2 = 8</math>. It is parallel and congruent to <math>AB</math>, so the width of the rectangle is <math>8</math>. | ||
− | Thus, the area of the rectangle is <math>lw = 4\cdot 8 = 32</math>, and the answer is <math>\boxed{ | + | Thus, the area of the rectangle is <math>lw = 4\cdot 8 = 32</math>, and the answer is <math>\boxed{C}</math>. |
==Solution 2== | ==Solution 2== | ||
Line 40: | Line 40: | ||
* [[AJHSME Problems and Solutions]] | * [[AJHSME Problems and Solutions]] | ||
* [[Mathematics competition resources]] | * [[Mathematics competition resources]] | ||
+ | {{MAA Notice}} |
Latest revision as of 00:15, 5 July 2013
Contents
Problem
Three congruent circles with centers ,
, and
are tangent to the sides of rectangle
as shown. The circle centered at
has diameter
and passes through points
and
. The area of the rectangle is
Solution 1
If circle has diameter
, then so do congruent circles
and
. Draw a diameter through
parallel to
. The diameter will be congruent to
, and thus
, which is the height of the rectangle.
Draw a horizontal line that extends to the sides of the rectangle. This line is
diameters long, so it has length
. It is parallel and congruent to
, so the width of the rectangle is
.
Thus, the area of the rectangle is , and the answer is
.
Solution 2
The area of circle and circle
almost fill the rectangle. Circle
has radius
, and so does circle
. Thus, the sum of their areas is
. Since
, the area of the circles is just over
. The area of the rectangle is greater than the area of the circles.
would be too high, as the two circles appear to take up much more than half the area of the rectangle. But
is too low, as it is less than the area of the two circles. Thus, the only reasonable answer is
, which is
.
See Also
1995 AJHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
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 AJHSME/AMC 8 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.