Difference between revisions of "1985 AHSME Problems/Problem 3"
Sevenoptimus (talk | contribs) m (Improved formatting) |
J314andrews (talk | contribs) (Shrunk diagram because it was huge.) |
||
| Line 4: | Line 4: | ||
<asy> | <asy> | ||
defaultpen(linewidth(0.7)+fontsize(10)); | defaultpen(linewidth(0.7)+fontsize(10)); | ||
| + | unitsize(0.5 cm); | ||
pair A=origin, B=(12,7), C=(12,0), M=12*dir(A--B), N=B+B.y*dir(B--A); | pair A=origin, B=(12,7), C=(12,0), M=12*dir(A--B), N=B+B.y*dir(B--A); | ||
real r=degrees(B); | real r=degrees(B); | ||
| Line 11: | Line 12: | ||
label("$B$", B, dir(point--B)); | label("$B$", B, dir(point--B)); | ||
label("$C$", C, dir(point--C)); | label("$C$", C, dir(point--C)); | ||
| − | label("$M$", M, | + | label("$M$", M, NW); |
| − | label("$N$", N, | + | label("$N$", N, NW); |
label("$12$", (6,0), S); | label("$12$", (6,0), S); | ||
label("$5$", (12,3.5), E);</asy> | label("$5$", (12,3.5), E);</asy> | ||
Latest revision as of 23:16, 3 July 2025
Problem
In right 
with legs 
and 
, arcs of circles are drawn, one with center 
and radius , the other with center 
and radius . They intersect the hypotenuse in 
and 
. Then 
has length
![[asy] defaultpen(linewidth(0.7)+fontsize(10)); unitsize(0.5 cm); pair A=origin, B=(12,7), C=(12,0), M=12*dir(A--B), N=B+B.y*dir(B--A); real r=degrees(B); draw(A--B--C--cycle^^Arc(A,12,0,r)^^Arc(B,B.y,180+r,270)); pair point=incenter(A,B,C); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$M$", M, NW); label("$N$", N, NW); label("$12$", (6,0), S); label("$5$", (12,3.5), E);[/asy]](http://latex.artofproblemsolving.com/1/1/5/1158a86e35b0a86d3be0efd981771a0f1a3d03c5.png)
Solution
Firstly, the Pythagorean theorem gives 
Also, 
and 
since they are both radii of the respective circles. Thus 
, and so 
.
See Also
| 1985 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 2 |
Followed by Problem 4 | |
| 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 | ||
| All AHSME Problems and Solutions | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
