ABCD is a parallelogram in which BC is produced to E such that CE = BC and AE intersects CD at F. If ar.(△ DFB) = 30 cm²; find the area of parallelogram.
Theorems on Area
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ABCD is a parallelogram in which BC is produced to E such that CE = BC and AE intersects CD at F. If ar.(△ DFB) = 30 cm²; find the area of parallelogram.

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We know that, Area of triangles on the same base and between the same parallel lines are equal. △ ADF and △ DFB lie on same base DF and between same parallel lines AB and DC. ∴ Area of △ ADF = Area of △ DFB = 30 cm² By converse of mid-point theorem, If a line is drawn through the midpoint of one side of a triangle, and parallel to the other side, it bisects the third side. In △ ABE, C is the mid-point of BE and CF || AB. ∴ F is the mid-point of AE. (By converse of mid-point theorem) ∴ EF = AF. In △ ADF and △ EFC, ⇒ ∠AFD = ∠EFC (Vertically opposite angles are equal) ⇒ EF = AF (Proved above) ⇒ ∠DAF = ∠CEF (Alternate interior angles are equal) ∴ △ ADF ≅ △ EFC (By A.S.A. axiom) We know that, Area of congruent triangles are equal. ∴ Area of △ EFC = Area of △ ADF = 30 cm². In △ BFE, Since, C is the mid-point of BE. ∴ CF is the mid-point of median of triangle. We know that, Median of triangle divides it into two triangles of equal areas. ∴ Area of △ BFC = Area of △ EFC = 30 cm². From figure, ⇒ Area of △ BDC = Area of △ BDF + Area of △ BFC ⇒ Area of △ BDC = 30 + 30 = 60 cm². We know that, The area of triangle is half that of a parallelogram on the same base and between the same parallels. From figure, || gm ABCD and △ BDC lies on same base DC and between same parallel lines AB and DC. ∴ Area of △ BDC = Area of || gm ABCD ⇒ Area of || gm ABCD = 2 × Area of △ BDC ⇒ Area of || gm ABCD = 2 × 60 = 120 cm². Hence, area of || gm ABCD = 120 cm².

Mathematics

ABCD is a parallelogram in which BC is produced to E such that CE = BC and AE intersects CD at F. If ar.(△ DFB) = 30 cm²; find the area of parallelogram.

Theorems on Area

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Mathematics

ABCD is a parallelogram in which BC is produced to E such that CE = BC and AE intersects CD at F. If ar.(△ DFB) = 30 cm2; find the area of parallelogram.

Theorems on Area

Related Questions

In the given figure, ABCD is a parallelogram. BC is produced to point X. Prove that :area (△ ABX) = area (quad.ACXD)

The given figure shows parallelograms ABCD and APQR. Show that these parallelograms are equal in area.

The following figure shows a triangle ABC in which P, Q and R are mid-points of sides AB, BC and CA respectively. S is mid-point of PQ. Prove that :ar.(△ ABC) = 8 × ar.(△ QSB)

In the given figure, the diagonals AC and BD intersect at point O. If OB = OD and AB // DC, prove that :(i) Area of (△ DOC) = Area of (△ AOB)(ii) Area of (△ DCB) = Area of (△ ACB)(iii) ABCD is a parallelogram.

ABCD is a parallelogram in which BC is produced to E such that CE = BC and AE intersects CD at F. If ar.(△ DFB) = 30 cm²; find the area of parallelogram.

In the given figure, ABCD is a parallelogram. BC is produced to point X. Prove that :
area (△ ABX) = area (quad.ACXD)

The following figure shows a triangle ABC in which P, Q and R are mid-points of sides AB, BC and CA respectively. S is mid-point of PQ. Prove that :
ar.(△ ABC) = 8 × ar.(△ QSB)

In the given figure, the diagonals AC and BD intersect at point O. If OB = OD and AB // DC, prove that :
(i) Area of (△ DOC) = Area of (△ AOB)
(ii) Area of (△ DCB) = Area of (△ ACB)
(iii) ABCD is a parallelogram.