Difference between revisions of "1984 AHSME Problems/Problem 4"
m (Created page with "==Problem== Points <math> B, C, F, E </math> are picked on a circle such that <math> BC||EF </math>. When <math> BC </math> is extended to the left, point <math> A </math> is...") |
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− | + | A rectangle intersects a circle as shown: <math>AB=4</math>, <math>BC=5</math>, and <math>DE=3</math>. Then <math>EF</math> equals: | |
− | |||
− | {{ | + | <asy>defaultpen(linewidth(0.7)+fontsize(10)); |
− | + | pair D=origin, E=(3,0), F=(10,0), G=(12,0), H=(12,1), A=(0,1), B=(4,1), C=(9,1), O=circumcenter(B,C,F); | |
+ | draw(D--G--H--A--cycle); | ||
+ | draw(Circle(O, abs(O-C))); | ||
+ | label("$A$", A, NW); | ||
+ | label("$B$", B, NW); | ||
+ | label("$C$", C, NE); | ||
+ | label("$D$", D, SW); | ||
+ | label("$E$", E, SE); | ||
+ | label("$F$", F, SW); | ||
+ | |||
+ | label("4", (2,0.85), N); | ||
+ | label("3", D--E, S); | ||
+ | label("5", (6.5,0.85), N); | ||
+ | </asy> | ||
+ | <math>\mathbf{(A)}\; 6\qquad \mathbf{(B)}\; 7\qquad \mathbf{(C)}\; \frac{20}3\qquad \mathbf{(D)}\; 8\qquad \mathbf{(E)}\; 9</math> | ||
==Solution== | ==Solution== |
Revision as of 16:36, 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 |