1964 AMC 12 Problem 17

Question

Given the distinct points

P(x_1,y_1),

Q(x_2,y_2)

and

R(x_1+x_2,y_1+y_2).

Line segments are drawn connecting these points to each other and to the origin

O.

Of the three possibilities:

(1)

parallelogram,

(2)

straight line,

(3)

trapezoid, figure

OPRQ,

depending upon the location of the points

P,

Q,

and

R,

can be:

(1)

only

(2)

only

(3)

only

(1)

or

(2)

only

all three

Answer: D

Concepts:vector parallelogram logical deduction

Difficulty rating: 1450

Small hint:

Interpret the position vectors as

R=P+Q

Big hint:

Separate the cases in which the vectors

P

and

Q

are independent or dependent

Solution:

The relation

\overrightarrow{OR}=\overrightarrow{OP}+\overrightarrow{OQ}

makes opposite sides of

OPRQ

parallel and equal whenever

P

and

Q

are not collinear, so the figure is a parallelogram. If the two vectors are dependent, all four points lie on a straight line. It cannot be a genuine trapezoid.

Thus, the correct answer is D.

1964 AMC 12 Problem 17

Answer: D

Difficulty rating: 1450

Small hint:

Big hint:

Solution:

Problem 17 in other years