In the given figure, AB = AD, BC = CD, BE = 2, BC = 4 and AC = 10.
What is the area of triangle ABD?

The correct answer is:

Area of a triangle is × base × height.
A kite is a type of quadrilateral that has 2 pairs of sides that are equal in lengths adjacent to each other.
The given ABCD is a kite since it has 2 pairs of sides i.e. AB & AD and BC & CD that are equal in lengths with each other (among the pair) and are adjacent to each other. The 2 diagonals of a kite intersect each other at a 90° angle and the longer diagonal bisects the shorter one into 2 halves.
Step-by-step solution:-
From the above explanations, we can conclude that-
i. For the triangle ABD, AE is the height which is perpendicular to BD and
ii. BE = DE = 2
∴ BD = BE + DE = 2 + 2 = 4 units --------------------------- (Equation i)
Now, for triangle EBC, we know that BE is perpendicular to CE, forming a 90° angle.
∴ we can apply pythagoras theorem.
∴ square of hypoteneous (BC) = sum of squares of the lengths of remaining 2 sides (BE & CE)
∴ (BC)² = (BE)² + (CE)²
∴ 4² = 2² + (CE)²
∴ 16 = 4 + (CE)²
∴ 16 - 4 = (CE)²
∴ 12 = (CE)²
∴ ± = CE
Since length of a side of a triangle cannot be negative,
CE = =  = .------------------- ( Equation ii)
AC = 10 ------------------------- (given)
From the given diagram, AC = AE + CE
∴ 10 = AE +  ------------------ (From Equation ii)
∴ 10 - = AE _------------------ (Equation iii)
For Area of triangle ABD-
Area = × height × base
= × AE × BD
= × (10- ) × 4 .-------------------------- (From equations i & iii)
= 2 × (10 - )
= 20 -
Final Answer:-
Option C i.e. 20 - is the correct Option.