Difference between revisions of "1984 AHSME Problems/Problem 4"
(→Problem) |
(→Solution) |
||
Line 20: | Line 20: | ||
==Solution== | ==Solution== | ||
<asy> | <asy> | ||
− | + | defaultpen(linewidth(0.7)+fontsize(10)); | |
− | + | pair D=origin, E=(3,0), F=(10,0), X=(12,0), Y=(12,1), A=(0,1), B=(4,1), C=(9,1), O=circumcenter(B,C,F), G=foot(E,A,C), H=foot(B,D,F), I=foot(C,D,F); | |
− | + | draw(D--X--Y--A--cycle); | |
− | + | draw(Circle(O, abs(O-C))); | |
− | + | label("$A$", A, NW); | |
− | + | label("$B$", B, N); | |
− | + | label("$C$", C, NE); | |
− | draw( | + | label("$D$", D, SW); |
− | + | label("$E$", E, S); | |
− | draw(( | + | label("$F$", F, S); |
− | label("$A$", | + | label("$G$", G, N); |
− | label("$B$", | + | label("$H$", H, S); |
− | label("$C$", | + | label("$I$", I, S); |
− | label("$D$", | + | |
− | label("$E$", | + | label("4", (2,0.85), N); |
− | label("$F$", | + | label("3", D--E, S); |
− | label("$G$", | + | label("5", (6.5,0.85), N); |
− | label("$H$", | + | draw(E--G^^H--B^^I--C, linetype("4 4")); |
− | label("$I$", | ||
− | label(" | ||
− | label(" | ||
− | label(" | ||
</asy> | </asy> | ||
Revision as of 16:39, 30 August 2011
A rectangle intersects a circle as shown: ,
, and
. Then
equals:
Solution
Draw and
, forming a trapezoid. Since it's cyclic, this trapezoid must be isosceles. Also, drop altitudes from
to
,
to
, and
to
, and let the feet of these altitudes be
,
, and
respectively.
is a rectangle since it has
right angles. Therefore,
, and
. By the same logic,
is also a rectangle, and
.
since they're both altitudes to a trapezoid, and
since the trapezoid is isosceles. Therefore, $\triangleBHE\congruent\triangleCIF$ (Error compiling LaTeX. Unknown error_msg) by HL congruence, so
. Also,
is a rectangle from
right angles, and
. Therefore,
.
See Also
1984 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
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 |