Question - Relationship Between Triangle Areas in a Quadrilateral

Let the area of triangle \( ABD \) be \( A_{ABD} \).

According to the problem, the area of quadrilateral \( ABCD \) is \( 6 \times A_{ABD} \)

The area of triangle \( ABC \) can be expressed as:

\( A_{ABC} = A_{ABD} + A_{BCD} \)

Since \( ABCD \) is a quadrilateral with right angles, triangle \( BCD \) is congruent to triangle \( ABD \). Thus, we can say:

\( A_{BCD} = A_{ABD} \)

Therefore, \( A_{ABC} = A_{ABD} + A_{ABD} = 2A_{ABD} \)

The ratio of the area of triangle \( ABC \) to the area of triangle \( ABD \) is:

\( \frac{A_{ABC}}{A_{ABD}} = \frac{2A_{ABD}}{A_{ABD}} = 2 \)

Hence, the answer is 2.