Difference between revisions of "1957 AHSME Problems/Problem 46"
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== Solution == | == Solution == | ||
| + | <asy> | ||
| + | |||
| + | import geometry; | ||
| + | |||
| + | point A = (0,0); | ||
| + | point B = (6,3); | ||
| + | point C = (8,0); | ||
| + | point D, P; | ||
| + | |||
| + | circle c = circumcircle(A,B,C); | ||
| + | |||
| + | // Triangle ABC w/ Circumcircle | ||
| + | draw(triangle(A,B,C)); | ||
| + | dot(A); | ||
| + | label("A",A,W); | ||
| + | dot(B); | ||
| + | label("B",B,N); | ||
| + | dot(C); | ||
| + | label("C",C,E); | ||
| + | draw(c); | ||
| + | |||
| + | // Segment BD, Triangle ADC | ||
| + | pair[] d = intersectionpoints(perpendicular(B,line(A,C)),c); | ||
| + | D = d[0]; | ||
| + | dot(D); | ||
| + | label("D",D,S); | ||
| + | draw(B--D); | ||
| + | draw(triangle(A,D,C)); | ||
| + | |||
| + | pair[] p = intersectionpoints(B--D,A--C); | ||
| + | P = p[0]; | ||
| + | dot(P); | ||
| + | label("P",P,SW); | ||
| + | |||
| + | // Right angle mark | ||
| + | markscalefactor = 0.0577; | ||
| + | draw(rightanglemark(A,P,B)); | ||
| + | |||
| + | // Length Labels | ||
| + | label("$3$", midpoint(B--P), W); | ||
| + | label("$4$", midpoint(P--D), W); | ||
| + | label("$6$", midpoint(A--P), S); | ||
| + | label("$2$", midpoint(P--C), S); | ||
| + | |||
| + | </asy> | ||
| + | |||
<math>\boxed{\textbf{(E) }\sqrt{65}}</math>. | <math>\boxed{\textbf{(E) }\sqrt{65}}</math>. | ||
Revision as of 12:31, 27 July 2024
Problem
Two perpendicular chords intersect in a circle. The segments of one chord are 
and 
;
the segments of the other are 
and 
. Then the diameter of the circle is:
Solution
![[asy] import geometry; point A = (0,0); point B = (6,3); point C = (8,0); point D, P; circle c = circumcircle(A,B,C); // Triangle ABC w/ Circumcircle draw(triangle(A,B,C)); dot(A); label("A",A,W); dot(B); label("B",B,N); dot(C); label("C",C,E); draw(c); // Segment BD, Triangle ADC pair[] d = intersectionpoints(perpendicular(B,line(A,C)),c); D = d[0]; dot(D); label("D",D,S); draw(B--D); draw(triangle(A,D,C)); pair[] p = intersectionpoints(B--D,A--C); P = p[0]; dot(P); label("P",P,SW); // Right angle mark markscalefactor = 0.0577; draw(rightanglemark(A,P,B)); // Length Labels label("$3$", midpoint(B--P), W); label("$4$", midpoint(P--D), W); label("$6$", midpoint(A--P), S); label("$2$", midpoint(P--C), S); [/asy]](http://latex.artofproblemsolving.com/8/b/d/8bd59c69fecdc4a648032b82fc852abf7137eb70.png)
.
See Also
| 1957 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 45 |
Followed by Problem 47 | |
| 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 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
