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MAA AMC
AMC 12/AHSME
1966 AMC 12/AHSME
32
32
Part of
1966 AMC 12/AHSME
Problems
(1)
Ratios
Source: 1966 AHSME #32
8/31/2011
Let
M
M
M
be the midpoint of side
A
B
AB
A
B
of the triangle
A
B
C
ABC
A
BC
. Let
P
P
P
be a point on
A
B
AB
A
B
between
A
A
A
and
M
M
M
, and let
M
D
MD
M
D
be drawn parallel to
P
C
PC
PC
and intersecting
B
C
BC
BC
at
D
D
D
. If the ratio of the area of the triangle
B
P
D
BPD
BP
D
to that of triangle
A
B
C
ABC
A
BC
is denoted by
r
r
r
, then
(A)
1
2
<
r
<
1
depending upon the position of
P
(B)
r
=
1
2
independent of the position of
P
(C)
1
2
≤
r
<
1
depending upon the position of
P
(D)
1
3
<
r
<
2
3
depending upon the position of
P
(E)
r
=
1
3
independent of the position of
P
\text{(A)}\ \tfrac{1}{2}<r<1\text{ depending upon the position of }P \qquad\\ \text{(B)}\ r=\tfrac{1}{2}\text{ independent of the position of }P\qquad\\ \text{(C)}\ \tfrac{1}{2}\le r<1\text{ depending upon the position of }P \qquad\\ \text{(D)}\ \tfrac{1}{3}<r<\tfrac{2}{3}\text{ depending upon the position of }P \qquad\\ \text{(E)}\ r=\tfrac{1}{3} \text{ independent of the position of }P
(A)
2
1
<
r
<
1
depending upon the position of
P
(B)
r
=
2
1
independent of the position of
P
(C)
2
1
≤
r
<
1
depending upon the position of
P
(D)
3
1
<
r
<
3
2
depending upon the position of
P
(E)
r
=
3
1
independent of the position of
P
ratio
geometry
AMC