Difference between revisions of "1983 AIME Problems/Problem 15"
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=== Solution 4 (coordinate geometry) === | === Solution 4 (coordinate geometry) === | ||
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Let the circle have equation <math>x^2 + y^2 = 25</math>, with centre <math>O(0,0)</math>. Since <math>BC=6</math>, we can calculate (by the Pythagorean Theorem) that the distance from <math>O</math> to the line <math>BC</math> is <math>4</math>. Therefore, we can let <math>B=(3,4)</math> and <math>C=(-3,4)</math>. Now, assume that <math>A</math> is any point on the major arc BC, and <math>D</math> any point on the minor arc BC. We can write <math>A=(5 \cos \alpha, 5 \sin \alpha)</math>, where <math>\alpha</math> is the angle measured from the positive <math>x</math> axis to the ray <math>OA</math>. It will also be convenient to define <math>\angle XOB = \alpha_0</math>. | Let the circle have equation <math>x^2 + y^2 = 25</math>, with centre <math>O(0,0)</math>. Since <math>BC=6</math>, we can calculate (by the Pythagorean Theorem) that the distance from <math>O</math> to the line <math>BC</math> is <math>4</math>. Therefore, we can let <math>B=(3,4)</math> and <math>C=(-3,4)</math>. Now, assume that <math>A</math> is any point on the major arc BC, and <math>D</math> any point on the minor arc BC. We can write <math>A=(5 \cos \alpha, 5 \sin \alpha)</math>, where <math>\alpha</math> is the angle measured from the positive <math>x</math> axis to the ray <math>OA</math>. It will also be convenient to define <math>\angle XOB = \alpha_0</math>. |
Revision as of 01:05, 7 February 2023
Contents
Problem
The adjoining figure shows two intersecting chords in a circle, with on minor arc
. Suppose that the radius of the circle is
, that
, and that
is bisected by
. Suppose further that
is the only chord starting at
which is bisected by
. It follows that the sine of the central angle of minor arc
is a rational number. If this number is expressed as a fraction
in lowest terms, what is the product
?
Solution
Solution 1
As with some of the other solutions, we analyze this with a locus—but a different one. We'll consider: given a point and a line
what is the set of points
such that the midpoint of
lies on line
? The answer to this question is: a line
parallel to
, such that
and
are (1) on opposite sides of
. and (2) at the same distance from
.
![[asy] size(170); pair P = (0,0), L1 = (-3, 0), L2 = (1.5, 1.5), M1 = (-3, 1), M2 = (1.5, 2.5); pair X = .6*M1 + .4*M2, M = (P+X)/2; draw(L1--L2); draw(M1--M2); draw(P--X, dotted); dot(M); dot("$P$", P, S); dot("$X$", X, N); label("$\ell$", L2, E); label("$m$", M2, E); [/asy]](http://latex.artofproblemsolving.com/a/b/c/abc1c6378210e4d7f4ca0c018733810d9d040ef6.png)
Applied to this problem, this means that is the only point that lies on both (1) the given circle, and (2) the line through
parallel to
. This means that
is parallel to the tangent to the given circle at
.
If we take to be the center of the given circle, then this means that
is perpendicular to
. Let
be the midpoint of chord
and let
be the intersection of
with the line through
parallel to
.
![[asy] size(170); pair O = (0,0), D = (0, 5), B = (-3, 4), C = (3, 4), A = (-4, 3), EE = (4, 3); pair M = (B+C)/2, NN = (A+EE)/2; draw(circle(O, 5)); draw(O--D); draw(B--C); draw(A--EE); draw(B--O--A); dot("$O$", O, SE); label("$D$", D, N); label("$B$", B, WNW); label("$A$", A, WNW); label("$C$", C, ENE); label("$M$", M, NE); label("$N$", NN, SE);[/asy]](http://latex.artofproblemsolving.com/5/d/5/5d5d77536f3faa364695106552b7cb3715021c86.png)
Since we know that
; since
(a radius of the circle) is 5, we can conclude that
is a 3-4-5 right triangle. Since
and line
are equidistant from line
we know that
, and thus
. This makes
also a 3-4-5 right triangle.
We're looking for , and we can find that using the angle subtraction formula for sine. We have
This is in lowest terms, so our answer is
.
Solution 2
-Credit to Adamz for diagram-
Let
be any fixed point on circle
, and let
be a chord of circle
. The locus of midpoints
of the chord
is a circle
, with diameter
. Generally, the circle
can intersect the chord
at two points, one point, or they may not have a point of intersection. By the problem condition, however, the circle
is tangent to
at point
.
Let be the midpoint of the chord
. From right triangle
, we have
. This gives
.
Notice that the distance equals
, where
is the radius of circle
.
Hence (where
represents the radius,
, of the large circle given in the question). Therefore, since
is clearly acute, we see that
Next, notice that . We can therefore apply the subtraction formula for
to obtain
It follows that
, such that the answer is
.
Solution 3
This solution, while similar to Solution 2, is arguably more motivated and less contrived.
Firstly, we note the statement in the problem that " is the only chord starting at
and bisected by
" – what is its significance? What is the criterion for this statement to be true?
We consider the locus of midpoints of the chords from . It is well-known that this is the circle with diameter
, where
is the center of the circle. The proof is simple: every midpoint of a chord is a dilation of the endpoint with scale factor
and center
. Thus, the locus is the result of the dilation with scale factor
and centre
of circle
. Let the center of this circle be
.
Now, is bisected by
if they cross at some point
on the circle. Moreover, since
is the only chord,
must be tangent to the circle
.
The rest of this problem is straightforward.
Our goal is to find , where
is the midpoint of
. We have
and
.
Let
be the projection of
onto
, and similarly let
be the projection of
onto
. Then it remains to find
so that we can use the addition formula for
.
As is a radius of circle
,
, and similarly,
. Since
, we have
. Thus
.
Further, we see that is a dilation of
about center
with scale factor
, so
.
Lastly, we apply the formula:
Thus the answer is
.
Solution 4 (coordinate geometry)
Let the circle have equation , with centre
. Since
, we can calculate (by the Pythagorean Theorem) that the distance from
to the line
is
. Therefore, we can let
and
. Now, assume that
is any point on the major arc BC, and
any point on the minor arc BC. We can write
, where
is the angle measured from the positive
axis to the ray
. It will also be convenient to define
.
Firstly, since must lie in the minor arc
, we see that
. However, since the midpoint of
must lie on
, and the highest possible
-coordinate of
is
, we see that the
-coordinate cannot be lower than
, that is,
.
Secondly, there is a theorem that says that, in a circle, if a chord is bisected by a radius, then they must be perpendicular. Therefore, suppose that is the intersection point of
and
, so that by the theorem,
is perpendicular to
. So, if
is the only chord starting at
which is bisected by
, this means that
is the only point on the chord
such that
is perpendicular to
. Now suppose that
, where
. The fact that
must be perpendicular to
is equivalent to the following equation:
which becomes
This rearranges to
Given that this equation must have only one real root , we study the following function:
First, by the fact that the equation has real solutions, its discriminant
must be non-negative, so we calculate
It is obvious that this is in fact non-negative. If it is actually zero, then , and
. In this case,
, so we have found a possible solution. We thus calculate
by the subtraction formula for
. This means that the answer is
.
Addendum to Solution 4
Since this is an AIME problem, we can assume that we are done since we have found one possible case. However, in a full-solution contest, we could not assume that the answer is unique, but would need to prove that this is the unique solution. This can be proven as follows.
Suppose that , which would mean that there could be two real roots of
, one lying in the interval
, and another outside of it. We also see, by Vieta's Formulas, that the average of the two roots is
, which is non-negative, so the root outside of
must be no less than
. By considering the graph of
, which is a "U-shaped" parabola, it is now evident that
and
. We can just use the second inequality:
so
The only way for this inequality to be satisfied is when (by applying the Cauchy-Schwarz inequality, or just plotting the line
to see that point
can't be above this line), which does not make sense in the original problem statement. (For it would mean that the point
lies in the half-plane above the line
, inclusive, and the half-plane below the line
, exclusive. This can be seen to be impossible by drawing the lines and observing that the intersection of the two half-planes does not share any point with the circle.)
Solution 5
Let the center of the circle be . Fix
and
. Then, as
moves around the circle, the locus of the midpoints of
is clearly a circle. Since the problems gives that
is the only chord starting at
bisected by
, it follows that the circle with diameter
and
is tangent to
.
Now, let the intersection of and
be
and let the midpoint of
(the center of the circle tangent to
that we described beforehand) be
. Drop the altitude from
to
and call its intersection with
. Drop the perpendicular from
to
and call its intersection with
. Clearly,
and since
is radius, it equals
. The same applies for
, which also equals
. By the Pythagorean theorem, we deduce that
, so
. This is very important information! Now we know that
, so by Power of a Point,
.
We’re almost there! Since by the Pythagorean theorem, , we deduce that
.
, so
. Furthermore, since
, we know that
. By the law of cosines,
Therefore,
. Now, drop the altitude from
to
and call its intersection with
. Then, by the Pythagorean theorem,
. Thus,
and
. As a result,
.
.
Solution 6
Let I be the intersection of AD and BC.
Lemma: if and only if
.
Proof: If AI = ID, we get AO = OD, and thus IO is a perpendicular bisector of AD. If , We can get
Let be this the circle with diameter AO.
Thus, we get , implying I must lie on
. I also must lie on BC. If both of these conditions are satisfied, The extension of AI intersecting with circle O will create a point D such that AI = ID. Thus, since AD is the only chord starting at A, there must be only 1 possible I, implying the circle with diameter AO is tangent to BC.
Now, create a coordinate system such that the origin is O and the y-axis is perpendicular to BC. The positive x direction is from B to C.
Let Z be (0,5).
Let Y be (-5,0).
Let X be the center of . Since
's radius is
, the altitude from X to OY is 1.5, since the altitude from I to OY is 4. XO is 2.5, so
.
. If we let
, we can find that what we are looking for is
, which we can evaluate and get
-Alexlikemath
Solution 7 (No Trig)
Let be the center of the circle. The locus of midpoints of chords with
as a endpoint is a circle with diameter
. Additionally, this circle must be tangent to
. Let the center of this circle be
. Let
be the midpoint of
,
be the foot of the perpendicular from
to
, and
be the foot of the perpendicular from
to
. Let
.
From right triangle , we get
. Thus,
.
Since ,
, and
is right,
. From quadrilateral
, we get
. Thus,
.
Since angles and
are right, we get
Thus,
.
~rayfish
See Also
1983 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Last question | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |