D, E and F are the mid-points of the sides BC, CA and AB respectively of triangle ABC. Prove that :
(i) BDEF is a parallelogram.
(ii) area of BDEF is half the area of △ABC.

Question

D, E and F are the mid-points of the sides BC, CA and AB respectively of triangle ABC. Prove that : (i) BDEF is a parallelogram. (ii) area of BDEF is half the area of △ABC.

KnowledgeBoat · Accepted Answer

(i) Given: ABC is a triangle. D, E and F are the mid-points of the sides BC, CA and AB respectively. To prove: BDEF is a parallelogram. Construction: Join DE, EF and FD. Proof: D, E and F are mid-points of sides BC, AC and AB respectively. Using mid-point theorem, So, DE ∥ BA and DE = BA ⇒ DE ∥ BF and DE = BF And, EF ∥ BC and EF = BC ⇒ EF ∥ BD and EF = BD. Hence, BDEF is a parallelogram. (ii) To prove: Ar.(∥gm BDEF) = Ar.(Δ ABC) Proof: From (i), BDEF is a parallelogram. D, E and F are mid-points of sides BC, AC and AB respectively. Using mid-point theorem, So, FD ∥ AC and FD = AC ⇒ FD ∥ EC and FD = EC And, EF ∥ BC and EF = BC ⇒ EF ∥ DC and EF = DC Hence, DCEF is a parallelogram. And, DE ∥ BA and DE = BA ⇒ DE ∥ AF and DE = AF And, DF ∥ AC and DF = AC ⇒ DF ∥ AE and DF = AE Hence, AFDE is a parallelogram. That means, AFDE, DCEF and BDEF all are parallelograms. Now, DF is a diagonal of parallelogram BDEF. Therefore, Ar.(Δ BDF) = Ar.(Δ DEF) ................(1) DE is a diagonal of parallelogram DCEF. So, Ar.(Δ DCE) = Ar.(Δ DEF) ................(2) FE is a diagonal of parallelogram AFDE. So, Ar.(Δ AFE) = Ar.(Δ DEF) ................(3) From (1), (2) and (3), we have Ar.(Δ BDF) = Ar.(Δ DCE) = Ar.(Δ AFE) = Ar.(Δ DEF) But, Ar.(Δ BDF) + Ar.(Δ DCE) + Ar.(Δ AFE) + Ar.(Δ DEF) = Ar.(Δ ABC) So, 4 x Ar.(Δ DEF) = Ar.(Δ ABC) ⇒ Ar.(Δ DEF) = Ar.(Δ ABC) We know that the diagonal of a parallelogram divides it into two triangles of equal areas. ⇒ Ar.(∥gm BDEF) = 2Ar.(Δ DEF) ⇒ Ar.(∥gm BDEF) = 2 x Ar.(Δ ABC) ⇒ Ar.(∥gm BDEF) = Ar.(Δ ABC) Hence, area of BDEF is half the area of △ABC.

Mathematics

D, E and F are the mid-points of the sides BC, CA and AB respectively of triangle ABC. Prove that :
(i) BDEF is a parallelogram.
(ii) area of BDEF is half the area of △ABC.

Theorems on Area

Related Questions

Construct a rhombus ABCD, such that each of its side is 4.8 cm and ∠A = 120°.

Construct a trapezium ABCD in which AB is parallel to DC, AB = 6.4 cm, ∠A = 60° and ∠B = 75°.

ABCD is a parallelogram of area 162 sq. cm. P is a point on AB such that AP : PB = 1 : 2. Calculate :
(i) the area of △APD
(ii) the ratio PA : DC.

Mathematics

D, E and F are the mid-points of the sides BC, CA and AB respectively of triangle ABC. Prove that :(i) BDEF is a parallelogram.(ii) area of BDEF is half the area of △ABC.

Theorems on Area

Related Questions

Construct a rhombus ABCD, such that each of its side is 4.8 cm and ∠A = 120°.

Construct a trapezium ABCD in which AB is parallel to DC, AB = 6.4 cm, ∠A = 60° and ∠B = 75°.

ABCD is a parallelogram of area 162 sq. cm. P is a point on AB such that AP : PB = 1 : 2. Calculate :(i) the area of △APD(ii) the ratio PA : DC.

D, E and F are the mid-points of the sides BC, CA and AB respectively of triangle ABC. Prove that :
(i) BDEF is a parallelogram.
(ii) area of BDEF is half the area of △ABC.

ABCD is a parallelogram of area 162 sq. cm. P is a point on AB such that AP : PB = 1 : 2. Calculate :
(i) the area of △APD
(ii) the ratio PA : DC.