If D, E, F are respectively the mid-points of the sides AB, BC and CA of an equilateral triangle ABC, prove that △DEF is also an equilateral triangle.
Mid-point Theorem
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Question

If D, E, F are respectively the mid-points of the sides AB, BC and CA of an equilateral triangle ABC, prove that △DEF is also an equilateral triangle.

KnowledgeBoat · Accepted Answer

Given, △ABC is an equilateral triangle. ⇒ AB = BC = AC By mid-point theorem, The line segment joining the mid-points of any two sides of a triangle is parallel to the third side and equal to half of it. Since, D and E are the mid-points of AB and BC respectively. ⇒ DE = × AC ⇒ DE = × AB [As AB = AC = BC] ....(1) Since, D and F are the mid-points of AB and AC respectively. ⇒ DF = × BC ⇒ DF = × AB [As AB = AC = BC] ....(2) Since, E and F are the mid-points of BC and AC respectively. ⇒ EF = × AB ....(3) From eq.(1), (2) and (3), we have: ⇒ DE = DF = EF ∴ △DEF is an equilateral triangle. Hence, proved that △DEF is an equilateral triangle.

Mathematics

If D, E, F are respectively the mid-points of the sides AB, BC and CA of an equilateral triangle ABC, prove that △DEF is also an equilateral triangle.

Mid-point Theorem

Related Questions

In the given figure, LMN is a right triangle in which ∠M = 90°, P and Q are mid-points of LM and LN respectively. If LM = 9 cm, MN = 12 cm and LN = 15 cm, find :
(i) the perimeter of trapezium MNQP
(ii) the area of trapezium MNQP

In the given figure, D, E, F are respectively the mid-points of the sides AB, BC and CA of △ABC. Prove that ADEF is a parallelogram.

In the adjoining figure, ABCD is a quadrilateral in which AD = BC and P, Q, R, S are the mid-points of AB, BD, CD and AC respectively. Prove that PQRS is a rhombus.

Mathematics

If D, E, F are respectively the mid-points of the sides AB, BC and CA of an equilateral triangle ABC, prove that △DEF is also an equilateral triangle.

Mid-point Theorem

Related Questions

In the given figure, LMN is a right triangle in which ∠M = 90°, P and Q are mid-points of LM and LN respectively. If LM = 9 cm, MN = 12 cm and LN = 15 cm, find :(i) the perimeter of trapezium MNQP(ii) the area of trapezium MNQP

In the given figure, D, E, F are respectively the mid-points of the sides AB, BC and CA of △ABC. Prove that ADEF is a parallelogram.

In the adjoining figure, ABCD is a quadrilateral in which AD = BC and P, Q, R, S are the mid-points of AB, BD, CD and AC respectively. Prove that PQRS is a rhombus.

In the given figure, LMN is a right triangle in which ∠M = 90°, P and Q are mid-points of LM and LN respectively. If LM = 9 cm, MN = 12 cm and LN = 15 cm, find :
(i) the perimeter of trapezium MNQP
(ii) the area of trapezium MNQP