Difference between revisions of "1995 AHSME Problems/Problem 26"
(New page: == Problem == In the figure, <math>\overline{AB}</math> and <math>\overline{CD}</math> are diameters of the circle with center <math>O</math>, <math>\overline{AB} \perp \overline{CD}</math...) |
m (→Problem: <asy> by dragon96) |
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== Problem == | == Problem == | ||
In the figure, <math>\overline{AB}</math> and <math>\overline{CD}</math> are diameters of the circle with center <math>O</math>, <math>\overline{AB} \perp \overline{CD}</math>, and chord <math>\overline{DF}</math> intersects <math>\overline{AB}</math> at <math>E</math>. If <math>DE = 6</math> and <math>EF = 2</math>, then the area of the circle is | In the figure, <math>\overline{AB}</math> and <math>\overline{CD}</math> are diameters of the circle with center <math>O</math>, <math>\overline{AB} \perp \overline{CD}</math>, and chord <math>\overline{DF}</math> intersects <math>\overline{AB}</math> at <math>E</math>. If <math>DE = 6</math> and <math>EF = 2</math>, then the area of the circle is | ||
| + | <!-- [[Image:1995 AHSME num.126.png]] --> | ||
| + | <asy>size(120); defaultpen(linewidth(0.7)); | ||
| + | pair O=origin, A=(-5,0), B=(5,0), C=(0,5), D=(0,-5), F=5*dir(40), E=intersectionpoint(A--B, F--D); | ||
| + | draw(Circle(O, 5)); | ||
| + | draw(A--B^^C--D--F); | ||
| + | dot(O^^A^^B^^C^^D^^E^^F); | ||
| + | markscalefactor=0.05; | ||
| + | draw(rightanglemark(B, O, D)); | ||
| − | + | label("$A$", A, dir(O--A)); | |
| − | + | label("$B$", B, dir(O--B)); | |
| − | [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=62477 Link to Image] | + | label("$C$", C, dir(O--C)); |
| − | + | label("$D$", D, dir(O--D)); | |
| + | label("$F$", F, dir(O--F)); | ||
| + | label("$O$", O, NW); | ||
| + | label("$E$", E, SE);</asy> | ||
| + | <!-- [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=62477 Link to Image] --> | ||
<math> \mathrm{(A) \ 23 \pi } \qquad \mathrm{(B) \ \frac {47}{2} \pi } \qquad \mathrm{(C) \ 24 \pi } \qquad \mathrm{(D) \ \frac {49}{2} \pi } \qquad \mathrm{(E) \ 25 \pi } </math> | <math> \mathrm{(A) \ 23 \pi } \qquad \mathrm{(B) \ \frac {47}{2} \pi } \qquad \mathrm{(C) \ 24 \pi } \qquad \mathrm{(D) \ \frac {49}{2} \pi } \qquad \mathrm{(E) \ 25 \pi } </math> | ||
Revision as of 20:26, 18 August 2011
Problem
In the figure, 
and 
are diameters of the circle with center 
, 
, and chord 
intersects at 
. If 
and 
, then the area of the circle is
![[asy]size(120); defaultpen(linewidth(0.7)); pair O=origin, A=(-5,0), B=(5,0), C=(0,5), D=(0,-5), F=5*dir(40), E=intersectionpoint(A--B, F--D); draw(Circle(O, 5)); draw(A--B^^C--D--F); dot(O^^A^^B^^C^^D^^E^^F); markscalefactor=0.05; draw(rightanglemark(B, O, D)); label("$A$", A, dir(O--A)); label("$B$", B, dir(O--B)); label("$C$", C, dir(O--C)); label("$D$", D, dir(O--D)); label("$F$", F, dir(O--F)); label("$O$", O, NW); label("$E$", E, SE);[/asy]](http://latex.artofproblemsolving.com/6/7/7/6778595cf613b7534e56d7a5a68ba8e859a5e9eb.png)
Solution
Let the radius of the circle be 
and let 
.
By the Pythagorean Theorem, 
.
By Power of a point, 
.
Adding these equations yields 
.
Thus, the area of the circle is 
.
See also
| 1995 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 25 |
Followed by Problem 27 | |
| 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 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
