Mathematics
In the given figure, D is mid-point of side AB of △ ABC and BDEC is a parallelogram.
Prove that :
Area of △ ABC = Area of // gm BDEC.
Answer
Given,
⇒ AD = DB ……..(1)
⇒ EC = DB (Opposite side of parallelogram are equal) ……..(2)
From equation (1) and (2), we get :
⇒ AD = EC
In △ EFC and △AFD,
⇒ ∠EFC = ∠AFD (Vertically opposite angles are equal)
⇒ AD = EC (Proved above)
⇒ ∠ECF = ∠FAD (Alternate angles are equal)
∴ △ EFC ≅ △ AFD (By A.A.S. axiom)
We know that,
Area of congruent triangles are equal.
∴ Area of △ EFC = Area of △ AFD
Adding area of quadrilateral CBDF on both sides of above equation, we get:
⇒ Area of △ EFC + Area of quad.CBDF = Area of △ AFD + Area of quad. CBDF
⇒ Area of || gm BDEC = Area of △ ABC.
Hence, proved that area of △ ABC = area of // gm BDEC.
Related Questions
ABCD and BCFE are parallelograms. If area of triangle EBC = 480 cm2, AB = 30 cm and BC = 40 cm; Calculate :
(i) area of parallelogram ABCD;
(ii) area of the parallelogram BCFE;
(iii) length of altitude from A on CD;
(iv) area of triangle ECF.
In the following figure, AC // PS // QR and PQ // DB // SR.
Prove that :
Area of quadrilateral PQRS = 2 × Area of quad.ABCD.
ABCD is a trapezium with AB // DC. A line parallel to AC intersects AB at point M and BC at point N. Prove that :
area of △ ADM = area of △ ACN.