Calculate the distance between points ${\text{B}}$ and ${\text{D}}$.

The distance between points ${\text{A}}$ and ${\text{C}}$ is $\sqrt {500}$.

Calculate the area of the kite ${\text{ABCD}}$.

${\text{BD}} = \sqrt {\left( {{4^2} + {8^2}} \right)}$     (M1)

Note:     Award (M1) for correct substitution into the distance formula.

$= 8.94\left( {8.94427 \ldots ,{\text{ }}\sqrt {80} ,{\text{ }}4\sqrt 5 } \right)$     (A1)     (C2)

Area ${\text{ABCD}} = 2 \times \left( {0.5 \times \frac{{{\text{their BD}}}}{2} \times \sqrt {500} } \right)$     (M1)(M1)(M1)

Note: Award (M1) for dividing their BD by 2, (M1) for correct substitution into the area of triangle formula, (M1) for adding two triangles (or multiplied by 2).

Accept alternative methods:

Area of kite $= 0.5 \times \sqrt {500}  \times$ their part (a).

Award (M1) for stating kite formula.

Award (M1) for correctly substituting in $\sqrt {500}$.

Award (M1) for correctly substituting in their part (a).

$= 100$     (A1)     (C4)

Note: Accept 99.9522 if 3 sf answer is used from part (a).