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Difference between revisions of "1975 AHSME Problems/Problem 11"

(→‎Solution)
(→‎Solution 1)
 
Line 12: Line 12:
 
==Solution 1==
 
==Solution 1==
 
It is (E) a circle.
 
It is (E) a circle.
 +
 +
The midpoints of the chord seems to be on the circle.
 +
For example, draw a chord passing through P.
 +
It is on the circle PK, where PK is the diameter.
 +
To prove this, we can use the fact that
 +
The perpendicular bisector of any chord of a circle passes through the center of the circle.
 +
 +
"Hence, the perpendicular through A passes through K.
 +
Since angle PAK=90, angle PAK is inscribed in the semicircular arc PK"
 +
 +
(Volume 2 , Chapter 20)
 +
 +
~Aarav22
  
 
==See Also==
 
==See Also==
 
{{AHSME box|year=1975|num-b=10|num-a=12}}
 
{{AHSME box|year=1975|num-b=10|num-a=12}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 11:14, 20 August 2025

Problem

Let $P$ be an interior point of circle $K$ other than the center of $K$. Form all chords of $K$ which pass through $P$, and determine their midpoints. The locus of these midpoints is

$\textbf{(A)} \text{ a circle with one point deleted} \qquad \\ \textbf{(B)} \text{ a circle if the distance from } P \text{ to the center of } K \text{ is less than one half the radius of } K; \\ \text{otherwise a circular arc of less than } 360^{\circ} \qquad \\ \textbf{(C)} \text{ a semicircle with one point deleted} \qquad \\ \textbf{(D)} \text{ a semicircle} \qquad \textbf{(E)} \text{ a circle}$

Solution 1

It is (E) a circle.

The midpoints of the chord seems to be on the circle. For example, draw a chord passing through P. It is on the circle PK, where PK is the diameter. To prove this, we can use the fact that The perpendicular bisector of any chord of a circle passes through the center of the circle.

"Hence, the perpendicular through A passes through K. Since angle PAK=90, angle PAK is inscribed in the semicircular arc PK"

(Volume 2 , Chapter 20)

~Aarav22

See Also

1975 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 10 		Followed by
Problem 12
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

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