Difference between revisions of "1983 AIME Problems/Problem 14"
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== Problem == | == Problem == | ||
In the adjoining figure, two circles with radii <math>8</math> and <math>6</math> are drawn with their centers <math>12</math> units apart. At <math>P</math>, one of the points of intersection, a line is drawn in such a way that the chords <math>QP</math> and <math>PR</math> have equal length. Find the square of the length of <math>QP</math>. | In the adjoining figure, two circles with radii <math>8</math> and <math>6</math> are drawn with their centers <math>12</math> units apart. At <math>P</math>, one of the points of intersection, a line is drawn in such a way that the chords <math>QP</math> and <math>PR</math> have equal length. Find the square of the length of <math>QP</math>. | ||
<asy>size(160); defaultpen(linewidth(.8pt)+fontsize(11pt)); dotfactor=3; pair O1=(0,0), O2=(12,0); path C1=Circle(O1,8), C2=Circle(O2,6); pair P=intersectionpoints(C1,C2)[0]; path C3=Circle(P,sqrt(130)); pair Q=intersectionpoints(C3,C1)[0]; pair R=intersectionpoints(C3,C2)[1]; draw(C1); draw(C2); draw(O2--O1); dot(O1); dot(O2); draw(Q--R); label("$Q$",Q,NW); label("$P$",P,1.5*dir(80)); label("$R$",R,NE); label("12",waypoint(O1--O2,0.4),S);</asy> | <asy>size(160); defaultpen(linewidth(.8pt)+fontsize(11pt)); dotfactor=3; pair O1=(0,0), O2=(12,0); path C1=Circle(O1,8), C2=Circle(O2,6); pair P=intersectionpoints(C1,C2)[0]; path C3=Circle(P,sqrt(130)); pair Q=intersectionpoints(C3,C1)[0]; pair R=intersectionpoints(C3,C2)[1]; draw(C1); draw(C2); draw(O2--O1); dot(O1); dot(O2); draw(Q--R); label("$Q$",Q,NW); label("$P$",P,1.5*dir(80)); label("$R$",R,NE); label("12",waypoint(O1--O2,0.4),S);</asy> | ||
− | |||
− | |||
== Solution == | == Solution == | ||
Line 49: | Line 49: | ||
Subtracting, <math>8y^2=28\Rightarrow y^2=\frac72\Rightarrow x^2=\frac{65}2\Rightarrow QP^2=4x^2=\boxed{130}</math>. | Subtracting, <math>8y^2=28\Rightarrow y^2=\frac72\Rightarrow x^2=\frac{65}2\Rightarrow QP^2=4x^2=\boxed{130}</math>. | ||
− | === Solution 3 === | + | === Solution 3 (trig bash)=== |
Let <math>QP=PR=x</math>. Angles <math>QPA</math>, <math>APB</math>, and <math>BPR</math> must add up to <math>180^{\circ}</math>. By the [[Law of Cosines]], <math>\angle APB=\cos^{-1}\left(\frac{{-11}}{24}\right)</math>. Also, angles <math>QPA</math> and <math>BPR</math> equal <math>\cos^{-1}\left(\frac{x}{16}\right)</math> and <math>\cos^{-1}\left(\frac{x}{12}\right)</math>. So we have <center><math>\cos^{-1}\left(\frac{x}{16}\right)+\cos^{-1}\left(\frac{{-11}}{24}\right)=180^{\circ}-\cos^{-1}\left(\frac{x}{12}\right).</math></center> Taking the cosine of both sides, and simplifying using the addition formula for <math>\cos</math> as well as the identity <math>\sin^{2}{x} + \cos^{2}{x} = 1</math>, gives <math>x^2=\boxed{130}</math>. | Let <math>QP=PR=x</math>. Angles <math>QPA</math>, <math>APB</math>, and <math>BPR</math> must add up to <math>180^{\circ}</math>. By the [[Law of Cosines]], <math>\angle APB=\cos^{-1}\left(\frac{{-11}}{24}\right)</math>. Also, angles <math>QPA</math> and <math>BPR</math> equal <math>\cos^{-1}\left(\frac{x}{16}\right)</math> and <math>\cos^{-1}\left(\frac{x}{12}\right)</math>. So we have <center><math>\cos^{-1}\left(\frac{x}{16}\right)+\cos^{-1}\left(\frac{{-11}}{24}\right)=180^{\circ}-\cos^{-1}\left(\frac{x}{12}\right).</math></center> Taking the cosine of both sides, and simplifying using the addition formula for <math>\cos</math> as well as the identity <math>\sin^{2}{x} + \cos^{2}{x} = 1</math>, gives <math>x^2=\boxed{130}</math>. | ||
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Thus by Power of a Point in the circle passing through <math>Q</math>, <math>R</math>, and <math>S</math>, we have <math>x \cdot 2x = 10 \cdot (10+16) = 260</math>, so <math>x^2 = \boxed{130}</math>. | Thus by Power of a Point in the circle passing through <math>Q</math>, <math>R</math>, and <math>S</math>, we have <math>x \cdot 2x = 10 \cdot (10+16) = 260</math>, so <math>x^2 = \boxed{130}</math>. | ||
+ | |||
+ | |||
+ | ===Solution 5 (Pythagorean Theorem and little algebraic manipulation)=== | ||
+ | <asy> size(0,5cm); pair a=(8,0),b=(20,0),t=(14,0),m=(9.72456,5.31401),n=(20.58055,1.77134),p=(15.15255,3.54268),q=(4.29657,7.08535),r=(26,0); draw(b--a--m--n--cycle); draw(p--t); draw(q--m); draw(n--r); draw(circumcircle(origin,q,p)); draw(circumcircle((14,0),p,r)); draw(rightanglemark(a,m,n,24)); label("$A$",a,S); label("$B$",b,S); label("$M$",m,NE); label("$N$",n,NE); label("$P$",p,N); label("$Q$",q,NW); label("$R$",r,E); label("$12$",(14,0),SW); label("$T$", t , NW); </asy> | ||
+ | Note that the midpoint of <math>AB</math> is <math>T.</math> Also, since <math>AM,NB</math> bisect <math>QP</math> and <math>PR,</math> respectively, <math>P</math> is the midpoint of <math>MN.</math> Thus, <math>AM+NB=2PT.</math> let <math>AM=a,BN=b.</math> This means that <math>a+b=2PT.</math> From the median formula, <math>PT=\sqrt{14}.</math> Thus, <math>a+b=2\sqrt{14}.</math> Also, since <math>MP=PN</math>, from the Pythagorean Theorem, <math>8^2-a^2=6^2-b^2\implies a^2-b^2=28.</math> Thus, <math>a-b=\frac{28}{2\sqrt{14}}=\sqrt{14}.</math> We conclude that <math>QP=MN=\sqrt{12^2-(a-b)^2}=\sqrt{130}\implies\boxed{130}.</math> | ||
+ | ~pinkpig | ||
+ | |||
+ | ===Solution 6 (Only simple geometry and algebra needed) === | ||
+ | Looking at Drawing 2 (by the way, we don't need point <math>R</math>), we set <math>AM=a</math> and <math>BN=b</math>, and the desired length<math>QP=x=PR</math>. We know that a radius perpendicular to a chord bisects the chord, so <math>MP=\frac{x}{2}</math> and <math>PN=\frac{x}{2}</math>. Draw line <math>AP</math> and <math>PB</math>, and we see that they are radii of Circles <math>A</math> and <math>B</math>, respectively. We can write the Pythagorean relationships <math>a^2+(\frac{x}{2})^2=8^2</math> for triangle <math>AMP</math> and <math>b^2+(\frac{x}{2})^2=6^2</math> for triangle <math>BNP</math>. We also translate segment <math>MN</math> down so that <math>N</math> coincides with <math>B</math>, and form another right traingle. From that triangle, you can see that the shorter leg is on the left side, having length <math>a-b</math>, the longer leg is the same as <math>MN=x</math>, and the hypotenuse is <math>AB=12</math>. We can write the Pythagorean relationship <math>(a-b)^2+x^2=12^2</math>. Solving the system of 3 unknowns and 3 equations (One of the best ways to do it is to solve for <math>a</math> in the first equation and <math>b</math> in the second equation, and substitute into the third equation, get an equation only in terms of <math>x</math>, and solve), you find that <math>x=\sqrt{130}</math>, so <math>x^2 = \boxed{130}</math>. | ||
+ | |||
+ | Solution by Kinglogic | ||
+ | |||
+ | ===Solution 7=== | ||
+ | |||
+ | The centers are collinear, you can prove it (but that is already given in the later section [Proof that R,A, and B are collinear]). Drop a perpendicular from <math>P</math> to the lines that the centers are on. You then have 2 separate segments, separated by the foot of the altitude of <math>P</math>. Call them <math>a</math> and <math>b</math> respectively. Call the measure of the foot of the altitude of <math>P</math> <math>h</math>. You then have 3 equations: | ||
+ | |||
+ | <cmath>(1)a+b=12</cmath> (this is given by the fact that the distance between the centers is 12. | ||
+ | |||
+ | <cmath>(2)a^2+h^2=64</cmath>. This is given by the fact that P is on the circle with radius 8. | ||
+ | |||
+ | <cmath>(3)b^2+h^2=36</cmath>. This is given by the fact that P is on the circle with radius 6. | ||
+ | |||
+ | Subtract (3) from (2) to get that <math>a^2-b^2=28</math>. As per (1), then you have <math>a-b=\frac{7}{3}</math> (4). Add (1) and (4) to get that <math>2a=\frac{43}{3}</math>. Then substitute into (1) to get <math>b=\frac{29}{6}</math>. Substitute either a or b into (2) or (3) to get <math>h=\sqrt{455}{6}</math>. Then to get <math>PQ=PR</math> it is just <math>\sqrt{(b+6)^2+h^2}=\sqrt{\frac{65^2}{6^2}+\frac{455}{6^2}}=\sqrt{\frac{4680}{36}}=\sqrt{130}</math>. | ||
+ | |||
+ | <math>PQ^2=\boxed{130}</math> | ||
+ | |||
+ | -dragoon | ||
===Full Proof that R, A, B are collinear === | ===Full Proof that R, A, B are collinear === | ||
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NOTE: Note this collinearity only follows from the fact that <math>6</math> is half of <math>12</math> in the problem statement. The collinearity is untrue in general. | NOTE: Note this collinearity only follows from the fact that <math>6</math> is half of <math>12</math> in the problem statement. The collinearity is untrue in general. | ||
+ | |||
+ | ===Solution 8 (Coordinate Bash)=== | ||
+ | We use coordinate geometry to approach this problem. Let the center of larger circle be the origin <math>O_1</math>, the smaller circle be <math>O_2</math>, and the x-axis be <math>O_1O_2</math>. Hence, we can get the two circle equations: <math>x^2+y^2 = 64</math> and <math>(x-12)^2+y^2=36</math>. | ||
+ | |||
+ | Let point <math>P</math> be <math>(a, b)</math>. Noting that it lies on both circles, we can plug the coordinates into both equations: | ||
+ | <center><math>a^2 +b^2 = 64</math></center> | ||
+ | <center><math>(a-12)^2+ b^2 \Rightarrow a^2-24a+144+b^2 = 64</math></center> | ||
+ | |||
+ | Substituting <math>a^2+b^2 = 64</math> into equation 2 and solving for <math>a</math>, we get <math>a = \frac{43}{6}</math>. | ||
+ | The problem asks us to find <math>QP^2</math>, which is congruent to <math>PR^2</math>. Using the distance formula for <math>P(a, b)</math> and <math>R(18, 0)</math> (by Solution 7's collinear proof), we get <math>PR^2 = (18-a)^2 +b^2</math>. Using <math>a^2+b^2 = 64</math>, we find that <math>b^2 = \frac{455}{36}</math>. Plugging the variables <math>a</math> and <math>b^2</math> in, we get <math>PR^2 = QP^2 = \boxed{130}</math> | ||
+ | ~SoilMilk | ||
+ | |||
+ | ===Solution 9 (basic solution) === | ||
+ | |||
+ | Let the center of the circle with radius <math>8</math> be <math>A,</math> and let the center of the one with radius <math>6</math> be <math>B.</math> Also, let <math>QP = PR = x.</math> Using law of cosines on triangle <math>APB,</math> we have that <math>\cos {APB} = \left(-\frac{{11}}{24}\right).</math> Angle chasing gives that <math>\angle{QAR} = \angle{APB},</math> so its cosines must be the same. Applying law of cosines again on triangle <math>QAR,</math> we have <math>\left(2x^2\right)=64+324-2(8)(18)\left(-\frac{11}{24}\right),</math> which gives that <math>x^2 = \boxed{130}</math> | ||
+ | |||
+ | ~happypi31415 | ||
== See Also == | == See Also == |
Latest revision as of 17:48, 27 May 2024
Contents
- 1 Problem
- 2 Solution
- 2.1 Solution 1
- 2.2 Solution 2 (Easiest)
- 2.3 Solution 3 (trig bash)
- 2.4 Solution 4 (quickest)
- 2.5 Solution 5 (Pythagorean Theorem and little algebraic manipulation)
- 2.6 Solution 6 (Only simple geometry and algebra needed)
- 2.7 Solution 7
- 2.8 Full Proof that R, A, B are collinear
- 2.9 Solution 8 (Coordinate Bash)
- 2.10 Solution 9 (basic solution)
- 3 See Also
Problem
In the adjoining figure, two circles with radii and
are drawn with their centers
units apart. At
, one of the points of intersection, a line is drawn in such a way that the chords
and
have equal length. Find the square of the length of
.
Solution
Note that some of these solutions assume that lies on the line connecting the centers, which is not true in general. It is true here only because the perpendicular from
passes through through the point where the line between the centers intersects the small circle. This fact can be derived from the application of the Midpoint Theorem to the trapezoid made by dropping perpendiculars from the centers onto
.
Solution 1
Firstly, notice that if we reflect over
, we get
. Since we know that
is on circle
and
is on circle
, we can reflect circle
over
to get another circle (centered at a new point
, and with radius
) that intersects circle
at
. The rest is just finding lengths, as follows.
Since is the midpoint of segment
,
is a median of
. Because we know
,
, and
, we can find the third side of the triangle using Stewart's Theorem or similar approaches. We get
. Now we have a kite
with
,
, and
, and all we need is the length of the other diagonal
. The easiest way it can be found is with the Pythagorean Theorem. Let
be the length of
. Then
![$\sqrt{36-x^2} + \sqrt{64-x^2} = \sqrt{56}.$](http://latex.artofproblemsolving.com/6/d/2/6d2150ebb9706e5687fb702d57b9724260acea68.png)
Solving this equation, we find that , so
Solution 2 (Easiest)
Draw additional lines as indicated. Note that since triangles and
are isosceles, the altitudes are also bisectors, so let
.
Since triangles
and
are similar. If we let
, we have
.
Applying the Pythagorean Theorem on triangle , we have
. Similarly, for triangle
, we have
.
Subtracting, .
Solution 3 (trig bash)
Let . Angles
,
, and
must add up to
. By the Law of Cosines,
. Also, angles
and
equal
and
. So we have
![$\cos^{-1}\left(\frac{x}{16}\right)+\cos^{-1}\left(\frac{{-11}}{24}\right)=180^{\circ}-\cos^{-1}\left(\frac{x}{12}\right).$](http://latex.artofproblemsolving.com/0/7/5/0754c4f6c7144ced3c35f0a8ea07c99e1639a6c9.png)
Taking the cosine of both sides, and simplifying using the addition formula for as well as the identity
, gives
.
Solution 4 (quickest)
Let . Extend the line containing the centers of the two circles to meet
, and to meet the other side of the large circle at a point
.
The part of this line from to the point nearest to
where it intersects the larger circle has length
.
The length of the diameter of the larger circle is
.
Thus by Power of a Point in the circle passing through ,
, and
, we have
, so
.
Solution 5 (Pythagorean Theorem and little algebraic manipulation)
Note that the midpoint of
is
Also, since
bisect
and
respectively,
is the midpoint of
Thus,
let
This means that
From the median formula,
Thus,
Also, since
, from the Pythagorean Theorem,
Thus,
We conclude that
~pinkpig
Solution 6 (Only simple geometry and algebra needed)
Looking at Drawing 2 (by the way, we don't need point ), we set
and
, and the desired length
. We know that a radius perpendicular to a chord bisects the chord, so
and
. Draw line
and
, and we see that they are radii of Circles
and
, respectively. We can write the Pythagorean relationships
for triangle
and
for triangle
. We also translate segment
down so that
coincides with
, and form another right traingle. From that triangle, you can see that the shorter leg is on the left side, having length
, the longer leg is the same as
, and the hypotenuse is
. We can write the Pythagorean relationship
. Solving the system of 3 unknowns and 3 equations (One of the best ways to do it is to solve for
in the first equation and
in the second equation, and substitute into the third equation, get an equation only in terms of
, and solve), you find that
, so
.
Solution by Kinglogic
Solution 7
The centers are collinear, you can prove it (but that is already given in the later section [Proof that R,A, and B are collinear]). Drop a perpendicular from to the lines that the centers are on. You then have 2 separate segments, separated by the foot of the altitude of
. Call them
and
respectively. Call the measure of the foot of the altitude of
. You then have 3 equations:
(this is given by the fact that the distance between the centers is 12.
. This is given by the fact that P is on the circle with radius 8.
. This is given by the fact that P is on the circle with radius 6.
Subtract (3) from (2) to get that . As per (1), then you have
(4). Add (1) and (4) to get that
. Then substitute into (1) to get
. Substitute either a or b into (2) or (3) to get
. Then to get
it is just
.
-dragoon
Full Proof that R, A, B are collinear
Let and
be the feet of the perpendicular from
to
and
to
respectively. It is well known that a perpendicular from the center of a circle to a chord of that circle bisects the chord, so
, since the problem told us
.
We will show that lies on
.
Let be the intersection of circle centered at
with
. Then
.
Let ' be the foot of the perpendicular from
to
. Then
is a midline (or midsegment) in trapezoid
, so
coincides with
(they are both supposed to be the midpoint of
). In other words, since
, then
.
Thus, subtends a
degree arc. So arc
in circle
is
, so
is a diameter, as desired. Thus
,
,
are collinear.
NOTE: Note this collinearity only follows from the fact that is half of
in the problem statement. The collinearity is untrue in general.
Solution 8 (Coordinate Bash)
We use coordinate geometry to approach this problem. Let the center of larger circle be the origin , the smaller circle be
, and the x-axis be
. Hence, we can get the two circle equations:
and
.
Let point be
. Noting that it lies on both circles, we can plug the coordinates into both equations:
![$a^2 +b^2 = 64$](http://latex.artofproblemsolving.com/b/6/e/b6e9aef6202a7b1ae929dda7e0e89fbb20a5b138.png)
![$(a-12)^2+ b^2 \Rightarrow a^2-24a+144+b^2 = 64$](http://latex.artofproblemsolving.com/6/e/c/6ecf48dc2adec4ee3358d3816285434087d66d88.png)
Substituting into equation 2 and solving for
, we get
.
The problem asks us to find
, which is congruent to
. Using the distance formula for
and
(by Solution 7's collinear proof), we get
. Using
, we find that
. Plugging the variables
and
in, we get
~SoilMilk
Solution 9 (basic solution)
Let the center of the circle with radius be
and let the center of the one with radius
be
Also, let
Using law of cosines on triangle
we have that
Angle chasing gives that
so its cosines must be the same. Applying law of cosines again on triangle
we have
which gives that
~happypi31415
See Also
1983 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |