Difference between revisions of "1953 AHSME Problems/Problem 49"
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− | + | ==Problem== | |
+ | |||
+ | The coordinates of <math>A,B</math> and <math>C</math> are <math>(5,5),(2,1)</math> and <math>(0,k)</math> respectively. | ||
+ | The value of <math>k</math> that makes <math>\overline{AC}+\overline{BC}</math> as small as possible is: | ||
+ | |||
+ | <math>\textbf{(A)}\ 3 \qquad | ||
+ | \textbf{(B)}\ 4\frac{1}{2} \qquad | ||
+ | \textbf{(C)}\ 3\frac{6}{7} \qquad | ||
+ | \textbf{(D)}\ 4\frac{5}{6}\qquad | ||
+ | \textbf{(E)}\ 2\frac{1}{7} </math> | ||
+ | |||
+ | ==Solution== | ||
k will be between 1 and 5 for AC+BC to be minimum. If we mirror A across the Y axis as A' (-5,5), the distance A'C+BC will be same as AC+BC. The minimum of A'C+BC will occur when C is on the straight line connecting A' and B, i.e., C lies on the line A'B. So, we can find the Y-intercept of the line connecting A'(-5,5) and B(2,1), which is 15/7 = 2 1/7. so, the answer is <math>\boxed{(E) 2 1/7}.</math> | k will be between 1 and 5 for AC+BC to be minimum. If we mirror A across the Y axis as A' (-5,5), the distance A'C+BC will be same as AC+BC. The minimum of A'C+BC will occur when C is on the straight line connecting A' and B, i.e., C lies on the line A'B. So, we can find the Y-intercept of the line connecting A'(-5,5) and B(2,1), which is 15/7 = 2 1/7. so, the answer is <math>\boxed{(E) 2 1/7}.</math> | ||
+ | |||
+ | ==See Also== | ||
+ | |||
+ | {{AHSME 50p box|year=1953|num-b=48|num-a=50}} | ||
+ | |||
+ | {{MAA Notice}} |
Revision as of 19:08, 17 February 2020
Problem
The coordinates of and are and respectively. The value of that makes as small as possible is:
Solution
k will be between 1 and 5 for AC+BC to be minimum. If we mirror A across the Y axis as A' (-5,5), the distance A'C+BC will be same as AC+BC. The minimum of A'C+BC will occur when C is on the straight line connecting A' and B, i.e., C lies on the line A'B. So, we can find the Y-intercept of the line connecting A'(-5,5) and B(2,1), which is 15/7 = 2 1/7. so, the answer is
See Also
1953 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 48 |
Followed by Problem 50 | |
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