1978 AMC 12 Problem 29

Question

Sides

AB,

BC,

CD,

and

DA,

respectively, of convex quadrilateral

ABCD

are extended past

B,

C,

D,

and

A

to points

B',

C',

D',

and

A'.

Also,

AB=BB'=6,

BC=CC'=7,

CD=DD'=8,

and

DA=AA'=9;

and the area of

ABCD

is

10.

The area of

A'B'C'D'

is

20

40

45

50

60

In vector notation,

A'=2A-D,

B'=2B-A,

and similarly for the other two vertices

Substitute these expressions into the cross-product area formula for a polygon

Using position vectors, the equal extensions give

\begin{aligned} A'&=2A-D,& B'&=2B-A,\\ C'&=2C-B,& D'&=2D-C. \end{aligned}

Substitution into the oriented polygon-area sum shows that the diagonal cross terms cancel:

\begin{aligned} &A'\mathbin{\times}B' +B'\mathbin{\times}C'\\ &\quad+C'\mathbin{\times}D' +D'\mathbin{\times}A'\\ &=5A\mathbin{\times}B +5B\mathbin{\times}C\\ &\quad+5C\mathbin{\times}D +5D\mathbin{\times}A. \end{aligned}

Hence the outer area is five times the original area, or

5\cdot10=50.

Therefore, the correct answer is D.

Problem 29 in other years