Difference between revisions of "1958 AHSME Problems/Problem 48"
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== Solution == | == Solution == | ||
*If somebody wants to draw a diagram or make this solution better, PLEASE do so. I cannot express how bad I am at this. | *If somebody wants to draw a diagram or make this solution better, PLEASE do so. I cannot express how bad I am at this. | ||
| − | If P is on A, then the length is 10, eliminating answer choice (B). | + | If <math>P</math> is on <math>A</math>, then the length is 10, eliminating answer choice <math>(B)</math>. |
| − | If P is equidistant from C and D, the length is <math>2\sqrt{1^2+5^2}=2\sqrt{26}>10</math>, eliminating (A) and (C). | + | If <math>P</math> is equidistant from <math>C</math> and <math>D</math>, the length is <math>2\sqrt{1^2+5^2}=2\sqrt{26}>10</math>, eliminating <math>(A)</math> and <math>(C)</math>. |
| − | If | + | If <math>\deltaCDP</math> is right, then <math>\angleCDP</math> is right or <math>\angleDCP</math> is right. Assume that <math>\angleDCP</math> is right. <math>\deltaAPB</math> is right, so <math>CP=\sqrt{4*6}=\sqrt{24}</math>. Then, <math>DP=\sqrt{28}</math>, so the length we are looking for is <math>\sqrt{24}+\sqrt{28}>10</math>, eliminating (D). |
Thus, our answer is (E). | Thus, our answer is (E). | ||
Revision as of 23:29, 31 December 2023
Problem
Diameter 
of a circle with center 
is 
units. 
is a point 
units from 
, and on . 
is a point units from 
, and on . 
is any point on the circle. Then the broken-line path from to to :
Solution
- If somebody wants to draw a diagram or make this solution better, PLEASE do so. I cannot express how bad I am at this.
If is on , then the length is 10, eliminating answer choice 
.
If is equidistant from and , the length is 
, eliminating 
and 
.
If $\deltaCDP$ (Error compiling LaTeX. Unknown error_msg) is right, then $\angleCDP$ (Error compiling LaTeX. Unknown error_msg) is right or $\angleDCP$ (Error compiling LaTeX. Unknown error_msg) is right. Assume that $\angleDCP$ (Error compiling LaTeX. Unknown error_msg) is right. $\deltaAPB$ (Error compiling LaTeX. Unknown error_msg) is right, so 
. Then, 
, so the length we are looking for is 
, eliminating (D).
Thus, our answer is (E).
See Also
| 1958 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 47 |
Followed by Problem 49 | |
| 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. 
