1985 AMC 12 Problem 17

Question

Diagonal

DB

of rectangle

ABCD

is divided into three segments of length

1

by parallel lines

L

and

L'

that pass through

A

and

C

and are perpendicular to

DB.

The area of

ABCD,

rounded to one decimal place, is

4.1

4.2

4.3

4.4

4.5

Let the rectangle’s side lengths be

w,h;

then its diagonal has length

3

Project the vertical and horizontal sides onto the diagonal to obtain

\frac{h^2}{3}=1

and

\frac{w^2}{3}=2

Let

D=(0,0),

B=(w,h),

A=(0,h)

and

C=(w,0).

Since

DB=3,

we have

w^2+h^2=9.

The projection of

DA

onto

DB

has length

\frac{h^2}{3},

and the first marked segment has length

1,

so

h^2=3.

Similarly, the projection of

DC

reaches the second division point, so

\frac{w^2}{3}=2

and

w^2=6.

Hence the area is

wh=\sqrt{18}=3\sqrt2\approx4.2426,

which rounds to

4.2.

Therefore the correct answer is B.

Problem 17 in other years