Difference between revisions of "1965 AHSME Problems/Problem 30"
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== Solution 1 == | == Solution 1 == | ||
+ | |||
+ | <asy> | ||
+ | |||
+ | path circ, hyp; | ||
+ | |||
+ | draw((0,16)--(0,0)--(8,0)--(0,16)); | ||
+ | dot((0,16)); | ||
+ | label("A", (0,16), NW); | ||
+ | dot((0,0)); | ||
+ | label("C", (0,0), SW); | ||
+ | dot((8,0)); | ||
+ | label("B", (8,0), SE); | ||
+ | |||
+ | dot((0,8)); | ||
+ | label("F",(0,8),W); | ||
+ | |||
+ | markscalefactor=0.1; | ||
+ | draw(rightanglemark((0,16),(0,0),(8,0))); | ||
+ | |||
+ | circ=circle((4,0),4); | ||
+ | draw(circ); | ||
+ | hyp=(0,16)--(8,0); | ||
+ | pair [] x=intersectionpoints(circ,hyp); | ||
+ | dot(x[1]); | ||
+ | label("D",x[1],NE); | ||
+ | draw(x[1]--(0,8)); | ||
+ | draw(x[1]--(0,0)); | ||
+ | |||
+ | </asy> | ||
+ | |||
We will prove every result except for <math>\fbox{B}</math>. | We will prove every result except for <math>\fbox{B}</math>. | ||
Line 23: | Line 53: | ||
{{AHSME 40p box|year=1965|num-b=29|num-a=31}} | {{AHSME 40p box|year=1965|num-b=29|num-a=31}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
+ | [[Category:Introductory Geometry Problems]] |
Latest revision as of 09:29, 19 July 2024
Contents
Problem
Let of right triangle
be the diameter of a circle intersecting hypotenuse
in
.
At
a tangent is drawn cutting leg
in
. This information is not sufficient to prove that
Solution 1
We will prove every result except for .
By Thales' Theorem, and so
.
and
are both tangents to the same circle, and hence equal. Let
. Then
, and so
. We also have
, which implies
. This means that
, so
indeed bisects
. We also know that
, hence
. And
as
.
Since all of the results except for are true, our answer is
.
Solution 2
It's easy to verify that always equals
. Since
changes depending on the sidelengths of the triangle, we cannot be certain that
. Hence our answer is
.
See Also
1965 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 29 |
Followed by Problem 31 | |
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 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.