In △ ABC, D, E and F are mid-points of sides AB, BC and AC respectively. Prove that AE and DF bisect each other.

Given: ABC is a triangle and D, E and F are mid-points of sides AB, BC and AC respectively. To prove: AE and DF bisect each other. Construction: Join AE and DF. Proof: Since D and F are the midpoints of AB and AC, by the midpoint theorem, DF ∥ BC and DF = BC. ⇒ BC = 2DF Now, since E is the midpoint of BC, we can write: ⇒ BE = EC ...................(1) Similarly, let G be the point of intersection of AE and DF. We need to show that G is the midpoint of both segments. From the Midpoint Theorem applied in Δ ABE and Δ ACE: BE = 2DG and EC = 2GF ⇒ 2DG = 2GF (from equation (1)) ⇒ DG = GF Thus, G is the mid-point of AE. Also, since G lies on AE and AG = GE, G is also the midpoint of AE. Hence, AE and DF bisect each other.

In triangle ABC, P is mid-point of AB and Q is mid-point of AC. If AB = 9.6 cm, BC = 11 cm and AC = 11.2 cm; find the perimeter of the trapezium PBCQ.

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In △ ABC, D is mid-point of AB and E is mid-point of BC. Calculate :
(i) DE, if AC = 6.4 cm,
(ii) ∠DEB, if ∠ACB = 63°.

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In a kite shaped figure ABCD, AB = AD and CB = CD. Points P, Q and R are mid-points of sides AB, BC and CD respectively.
Prove that :
(i) ∠PQR = 90°
(ii) Line through P and parallel to QR bisects side AD.

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P, Q, R and S are the mid points of sides AB, BC, CD and DA respectively of rhombus ABCD. Show that PQRS is a rectangle.
Under what condition will PQRS be a square ?

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Mathematics

In △ ABC, D, E and F are mid-points of sides AB, BC and AC respectively. Prove that AE and DF bisect each other.

Mid-point Theorem

Related Questions

In triangle ABC, P is mid-point of AB and Q is mid-point of AC. If AB = 9.6 cm, BC = 11 cm and AC = 11.2 cm; find the perimeter of the trapezium PBCQ.

In △ ABC, D is mid-point of AB and E is mid-point of BC. Calculate :
(i) DE, if AC = 6.4 cm,
(ii) ∠DEB, if ∠ACB = 63°.

In a kite shaped figure ABCD, AB = AD and CB = CD. Points P, Q and R are mid-points of sides AB, BC and CD respectively.
Prove that :
(i) ∠PQR = 90°
(ii) Line through P and parallel to QR bisects side AD.

P, Q, R and S are the mid points of sides AB, BC, CD and DA respectively of rhombus ABCD. Show that PQRS is a rectangle.
Under what condition will PQRS be a square ?

Mathematics

In △ ABC, D, E and F are mid-points of sides AB, BC and AC respectively. Prove that AE and DF bisect each other.

Mid-point Theorem

Related Questions

In triangle ABC, P is mid-point of AB and Q is mid-point of AC. If AB = 9.6 cm, BC = 11 cm and AC = 11.2 cm; find the perimeter of the trapezium PBCQ.

In △ ABC, D is mid-point of AB and E is mid-point of BC. Calculate :(i) DE, if AC = 6.4 cm,(ii) ∠DEB, if ∠ACB = 63°.

In a kite shaped figure ABCD, AB = AD and CB = CD. Points P, Q and R are mid-points of sides AB, BC and CD respectively.Prove that :(i) ∠PQR = 90°(ii) Line through P and parallel to QR bisects side AD.

P, Q, R and S are the mid points of sides AB, BC, CD and DA respectively of rhombus ABCD. Show that PQRS is a rectangle.Under what condition will PQRS be a square ?

In △ ABC, D is mid-point of AB and E is mid-point of BC. Calculate :
(i) DE, if AC = 6.4 cm,
(ii) ∠DEB, if ∠ACB = 63°.

In a kite shaped figure ABCD, AB = AD and CB = CD. Points P, Q and R are mid-points of sides AB, BC and CD respectively.
Prove that :
(i) ∠PQR = 90°
(ii) Line through P and parallel to QR bisects side AD.

P, Q, R and S are the mid points of sides AB, BC, CD and DA respectively of rhombus ABCD. Show that PQRS is a rectangle.
Under what condition will PQRS be a square ?