Difference between revisions of "2001 AIME I Problems/Problem 7"
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== Solution == | == Solution == | ||
=== Solution 1 === | === Solution 1 === | ||
+ | <center><asy> | ||
+ | pointpen = black; pathpen = black+linewidth(0.7); | ||
+ | pair B=(0,0), C=(20,0), A=IP(CR(B,21),CR(C,22)), I=incenter(A,B,C), D=IP((0,I.y)--(20,I.y),A--B), E=IP((0,I.y)--(20,I.y),A--C); | ||
+ | D(MP("A",A,N)--MP("B",B)--MP("C",C)--cycle); D(MP("I",I,NE)); D(MP("E",E,NE)--MP("D",D,NW)); | ||
+ | // D((A.x,0)--A,linetype("4 4")+linewidth(0.7)); D((I.x,0)--I,linetype("4 4")+linewidth(0.7)); D(rightanglemark(B,(A.x,0),A,30)); | ||
+ | D(B--I--C); | ||
+ | MP("20",(B+C)/2); MP("21",(A+B)/2,NW); MP("22",(A+C)/2,NE); | ||
+ | </asy></center> | ||
+ | |||
+ | Let <math>I</math> be the [[incenter]] of <math>\triangle ABC</math>, so that <math>BI</math> and <math>CI</math> are [[angle bisector]]s of <math>\angle ABC</math> and <math>\angle ACB</math> respectively. Then, <math>\angle BID = \angle CBI = \angle DBI,</math> so <math>\triangle BDI</math> is [[Isosceles triangle|isosceles]], and similarly <math>\triangle CEI</math> is isosceles. It follows that <math>DE = DB + EC</math>, so the perimeter of <math>\triangle ADE</math> is <math>AD + AE + DE = AB + AC = 43</math>. Hence, the ratio of the perimeters of <math>\triangle ADE</math> and <math>\triangle ABC</math> is <math>\frac{43}{63}</math>, which is the scale factor between the two similar triangles, and thus <math>DE = \frac{43}{63} \times 20 = \frac{860}{63}</math>. Thus, <math>m + n = \boxed{923}</math>. | ||
+ | |||
+ | === Solution 2 === | ||
<center><asy> | <center><asy> | ||
pointpen = black; pathpen = black+linewidth(0.7); | pointpen = black; pathpen = black+linewidth(0.7); | ||
Line 24: | Line 36: | ||
Same as above, but simpler | Same as above, but simpler | ||
− | === Solution | + | === Solution 3 ([[mass points]]) === |
<center><asy> | <center><asy> | ||
pointpen = black; pathpen = black+linewidth(0.7); pen d = linewidth(0.7) + linetype("4 4"); | pointpen = black; pathpen = black+linewidth(0.7); pen d = linewidth(0.7) + linetype("4 4"); |
Revision as of 17:16, 5 August 2011
Problem
Triangle has
,
and
. Points
and
are located on
and
, respectively, such that
is parallel to
and contains the center of the inscribed circle of triangle
. Then
, where
and
are relatively prime positive integers. Find
.
Contents
[hide]Solution
Solution 1
![[asy] pointpen = black; pathpen = black+linewidth(0.7); pair B=(0,0), C=(20,0), A=IP(CR(B,21),CR(C,22)), I=incenter(A,B,C), D=IP((0,I.y)--(20,I.y),A--B), E=IP((0,I.y)--(20,I.y),A--C); D(MP("A",A,N)--MP("B",B)--MP("C",C)--cycle); D(MP("I",I,NE)); D(MP("E",E,NE)--MP("D",D,NW)); // D((A.x,0)--A,linetype("4 4")+linewidth(0.7)); D((I.x,0)--I,linetype("4 4")+linewidth(0.7)); D(rightanglemark(B,(A.x,0),A,30)); D(B--I--C); MP("20",(B+C)/2); MP("21",(A+B)/2,NW); MP("22",(A+C)/2,NE); [/asy]](http://latex.artofproblemsolving.com/a/e/4/ae47e3711da65c309f6b7dc3c0e55287b8c8a5fc.png)
Let be the incenter of
, so that
and
are angle bisectors of
and
respectively. Then,
so
is isosceles, and similarly
is isosceles. It follows that
, so the perimeter of
is
. Hence, the ratio of the perimeters of
and
is
, which is the scale factor between the two similar triangles, and thus
. Thus,
.
Solution 2
![[asy] pointpen = black; pathpen = black+linewidth(0.7); pair B=(0,0), C=(20,0), A=IP(CR(B,21),CR(C,22)), I=incenter(A,B,C), D=IP((0,I.y)--(20,I.y),A--B), E=IP((0,I.y)--(20,I.y),A--C); D(MP("A",A,N)--MP("B",B)--MP("C",C)--cycle); D(incircle(A,B,C)); D(MP("I",I,NE)); D(MP("E",E,NE)--MP("D",D,NW)); D((A.x,0)--A,linetype("4 4")+linewidth(0.7)); D((I.x,0)--I,linetype("4 4")+linewidth(0.7)); D(rightanglemark(B,(A.x,0),A,30)); MP("20",(B+C)/2); MP("21",(A+B)/2,NW); MP("22",(A+C)/2,NE); [/asy]](http://latex.artofproblemsolving.com/3/0/8/308015d62aeb9917bd2a032c366423e1e0529c1f.png)
The semiperimeter of is
. By Heron's formula, the area of the whole triangle is
. Using the formula
, we find that the inradius is
. Since
, the ratio of the heights of triangles
and
is equal to the ratio between sides
and
. From
, we find
. Thus, we have

Solving for gives
so the answer is
.
Or we have the area of the triangle as .
Using the ratio of heights to ratio of bases of
and
from that it is easy to deduce that
Same as above, but simpler
Solution 3 (mass points)
![[asy] pointpen = black; pathpen = black+linewidth(0.7); pen d = linewidth(0.7) + linetype("4 4"); pair B=(0,0), C=(20,0), A=IP(CR(B,21),CR(C,22)), I=incenter(A,B,C), D=IP((0,I.y)--(20,I.y),A--B), E=IP((0,I.y)--(20,I.y),A--C); D(MP("A",A,N)--MP("B",B)--MP("C",C)--cycle); D(incircle(A,B,C)); D(MP("P",I,(1,2))); D(MP("E",E,NE)--MP("D",D,NW)); MP("20",(B+C)/2); MP("21",(A+B)/2,NW); MP("22",(A+C)/2,NE); /* construct angle bisectors */ path anglebisector (pair X, pair Y, pair Z, path K) { return Y -- IP(Y -- Y + 30 * (bisectorpoint(X,Y,Z)-Y) , K); } D(anglebisector(C,A,B,B--C), d); D(anglebisector(B,C,A,A--B),d); D(anglebisector(C,B,A,A--C),d); [/asy]](http://latex.artofproblemsolving.com/7/c/7/7c752cc350ded25b2c947d336d77bc6ccf240f96.png)
Let be the incircle; then it is be the intersection of all three angle bisectors. Draw the bisector
to where it intersects
, and name the intersection
.
Using the angle bisector theorem, we know the ratio is
, thus we shall assign a weight of
to point
and a weight of
to point
, giving
a weight of
. In the same manner, using another bisector, we find that
has a weight of
. So, now we know
has a weight of
, and the ratio of
is
. Therefore, the smaller similar triangle
is
the height of the original triangle
. So,
is
the size of
. Multiplying this ratio by the length of
, we find
is
. Therefore,
.
See also
2001 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |