In the figure given below, AF = DF and AB // FE // DC. Prove that : (i) (ii) AB + CD = 2 EF

(i) Given: AF = DF and AB // FE // DC. To prove: Proof: In Δ FDP and Δ ADB, ∠ADB = ∠FDP (Common angle) ∠DAB = ∠DFP (corresponding angles since AB // FE) ∠DBA = ∠DPF (corresponding angles since AB // FE) By AAA similarity rule, Δ FDP ~ Δ ADB So, As it is given that AF = FD, FD = AD So, Hence, . (ii) To prove: AB + CD = 2 EF Proof: From (i), ⇒ 2FP = AB ......................(1) In Δ BEP and Δ DCB, ∠DBC = ∠PAE (Common angle) ∠BPE = ∠BDC (corresponding angles since AB // FE) ∠BEP = ∠BCD (corresponding angles since AB // FE) By AAA similarity rule, Δ BEP ~ Δ DCB So, As it is given that AF = DF and AB // FE // DC, so BE = EC BE = BC So, ⇒ 2EP = DC ......................(2) Adding equation (1) and (2), we get ⇒ 2FP + 2EP = AB + DC ⇒ 2(FP + EP) = AB + DC ⇒ 2FE = AB + DC Hence, AB + CD = 2 EF .

In quadrilateral ABCD, side DC is largest. Show that AB + AD > DC - BC.

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P is the mid-point of the hypotenuse AB of the right-angled triangle ABC. Prove that : AB = 2 CP.

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In △ABC, AB = AC. D, E and F are mid-points of the sides BC, CA and AB respectively. Show that :
(i) AD is perpendicular to FE.
(ii) AD and FE bisect each other.

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Mathematics

In the figure given below, AF = DF and AB // FE // DC. Prove that : (i) (ii) AB + CD = 2 EF

Mid-point Theorem

Related Questions

In quadrilateral ABCD, side DC is largest. Show that AB + AD > DC - BC.

P is the mid-point of the hypotenuse AB of the right-angled triangle ABC. Prove that : AB = 2 CP.

In △ABC, AB = AC. D, E and F are mid-points of the sides BC, CA and AB respectively. Show that :
(i) AD is perpendicular to FE.
(ii) AD and FE bisect each other.

Mathematics

In the figure given below, AF = DF and AB // FE // DC. Prove that : (i) (ii) AB + CD = 2 EF

Mid-point Theorem

Related Questions

In quadrilateral ABCD, side DC is largest. Show that AB + AD > DC - BC.

P is the mid-point of the hypotenuse AB of the right-angled triangle ABC. Prove that : AB = 2 CP.

In △ABC, AB = AC. D, E and F are mid-points of the sides BC, CA and AB respectively. Show that :(i) AD is perpendicular to FE.(ii) AD and FE bisect each other.

In △ABC, AB = AC. D, E and F are mid-points of the sides BC, CA and AB respectively. Show that :
(i) AD is perpendicular to FE.
(ii) AD and FE bisect each other.