Question

Points A(1,0), B(5,0), C(7,6) and D(3,6) are joined to form a quadrilateral ABCD. ABCD is a parallelogram with BD as its diagonal joined. BC is extended such that BC = BE. AE is joined. Which of the following is true?

• A. Area of ΔADE = (1/2) × Area of ΔBDC
• B. Area of ΔADE = (2) × Area of ΔBDC
• C. Area of ΔADE = (1) × Area of ΔBDC
• D. Area of ΔADE < Area of ΔBDC

Answer 1

The correct option is C

Area of ΔADE = (1) × Area of ΔBDC

ABCD is a quadrilateral with AB = CD and AB$\parallel$ CD. A quadrilateral with opposite sides parallel and equal is a parallelogram.
Since BC = BE, AD = BC. Thus, AD = BE. And CB is extended to BE, then BE is parallel to AD.
Since BE = AD, and BE ll AD, then ADBE is a parallelogram.
BD is a diagonal of the parallelogram ABCD. DE and AB are the diagonals of ADBE.
Triangles between the same base and same parallels are equal.
Area of $\Delta$ ADE = Area of $\Delta$ ABE.

Diagonals of parallelogram divides it into two triangles of equal area.
Area of $\Delta$ ABE = Area of $\Delta$ ADB (In parallelogram ADBE)

Thus, Area of $\Delta$ ADE = Area of $\Delta$ ADB.
Also, Area of$\Delta$ ADB = Area of $\Delta$ BDC (In parallelogram ABCD)

Hence, Area of $\Delta$ ADE = Area of $\Delta$BDC.