Difference between revisions of "2019 AIME II Problems/Problem 7"
(→Diagram: ; renamed and labeled points) |
MRENTHUSIASM (talk | contribs) (I decided to undo: The changes in the diagram section SHOULD NOT label more than the problem statement says. We should leave for the readers to assign points.) |
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Line 7: | Line 7: | ||
size(350); | size(350); | ||
− | pair A, B, C, D, | + | pair A, B, C, D, E, F, G, H, I, J, K, L; |
B = origin; | B = origin; | ||
C = (220,0); | C = (220,0); | ||
A = intersectionpoints(Circle(B,120),Circle(C,180))[0]; | A = intersectionpoints(Circle(B,120),Circle(C,180))[0]; | ||
− | D = | + | D = A+1/4*(B-A); |
− | + | E = A+1/4*(C-A); | |
− | F = | + | F = B+1/4*(A-B); |
− | G = | + | G = B+1/4*(C-B); |
− | H = | + | H = C+1/8*(A-C); |
− | I = | + | I = C+1/8*(B-C); |
− | + | J = extension(D,E,F,G); | |
− | + | K = extension(F,G,H,I); | |
− | + | L = extension(H,I,D,E); | |
draw(A--B--C--cycle); | draw(A--B--C--cycle); | ||
− | draw( | + | draw(J+9/8*(K-J)--K+9/8*(J-K),dashed); |
− | draw( | + | draw(L+9/8*(K-L)--K+9/8*(L-K),dashed); |
− | draw( | + | draw(J+9/8*(L-J)--L+9/8*(J-L),dashed); |
− | draw( | + | draw(D--E^^F--G^^H--I,red); |
dot("$B$",B,1.5SW,linewidth(4)); | dot("$B$",B,1.5SW,linewidth(4)); | ||
dot("$C$",C,1.5SE,linewidth(4)); | dot("$C$",C,1.5SE,linewidth(4)); | ||
dot("$A$",A,1.5N,linewidth(4)); | dot("$A$",A,1.5N,linewidth(4)); | ||
− | dot( | + | dot(D,linewidth(4)); |
− | dot( | + | dot(E,linewidth(4)); |
− | dot( | + | dot(F,linewidth(4)); |
− | dot( | + | dot(G,linewidth(4)); |
− | dot( | + | dot(H,linewidth(4)); |
− | dot( | + | dot(I,linewidth(4)); |
− | dot( | + | dot(J,linewidth(4)); |
− | dot( | + | dot(K,linewidth(4)); |
− | dot( | + | dot(L,linewidth(4)); |
− | label("$55$",midpoint( | + | label("$55$",midpoint(D--E),S,red); |
− | label("$ | + | label("$45$",midpoint(F--G),dir(55),red); |
− | label("$ | + | label("$15$",midpoint(H--I),dir(160),red); |
− | label("$\ell_A$", | + | label("$\ell_A$",J+9/8*(L-J),1.5*dir(B--C)); |
− | label("$\ell_B$", | + | label("$\ell_B$",K+9/8*(J-K),1.5*dir(C--A)); |
− | label("$\ell_C$", | + | label("$\ell_C$",L+9/8*(K-L),1.5*dir(A--B)); |
</asy> | </asy> | ||
− | ~MRENTHUSIASM | + | ~MRENTHUSIASM |
==Solution 1== | ==Solution 1== |
Revision as of 22:25, 4 November 2022
Problem
Triangle has side lengths
, and
. Lines
, and
are drawn parallel to
, and
, respectively, such that the intersections of
, and
with the interior of
are segments of lengths
, and
, respectively. Find the perimeter of the triangle whose sides lie on lines
, and
.
Diagram
~MRENTHUSIASM
Solution 1
Let the points of intersection of with
divide the sides into consecutive segments
. Furthermore, let the desired triangle be
, with
closest to side
,
closest to side
, and
closest to side
. Hence, the desired perimeter is
since
,
, and
.
Note that , so using similar triangle ratios, we find that
,
,
, and
.
We also notice that and
. Using similar triangles, we get that
Hence, the desired perimeter is
-ktong
Solution 2
Let the diagram be set up like that in Solution 1.
By similar triangles we have
Thus
Since and
, the altitude of
from
is half the altitude of
from
, say
. Also since
, the distance from
to
is
. Therefore the altitude of
from
is
.
By triangle scaling, the perimeter of is
of that of
, or
~ Nafer
Solution 3
Notation shown on diagram. By similar triangles we have
So,
vladimir.shelomovskii@gmail.com, vvsss
See Also
2019 AIME II (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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.