MathDB

In the diagram,

OB_i

is parallel and equal in length to

A_i A_{i + 1}

for

i = 1

, 2, 3, and 4 (

A_5 = A_1

). Show that the area of

B_1 B_2 B_3 B_4

is twice that of

A_1 A_2 A_3 A_4

.
[asy] unitsize(1 cm);
pair O; pair[] A, B;
O = (0,0); A[1] = (0.5,-3); A[2] = (2,0); A[3] = (-0.2,0.5); A[4] = (-1,0); B[1] = A[2] - A[1]; B[2] = A[3] - A[2]; B[3] = A[4] - A[3]; B[4] = A[1] - A[4];
draw(A[1]--A[2]--A[3]--A[4]--cycle); draw(B[1]--B[2]--B[3]--B[4]--cycle); draw(O--B[1]); draw(O--B[2]); draw(O--B[3]); draw(O--B[4]);
label("

A_1

", A[1], S); label("

A_2

", A[2], E); label("

A_3

", A[3], N); label("

A_4

", A[4], W); label("

B_1

", B[1], NE); label("

B_2

", B[2], W); label("

B_3

", B[3], SW); label("

B_4

", B[4], S); label("

O

", O, E); [/asy]

Problem Statement