Advertisements
Advertisements
Question
D, E and F are the mid-points of the sides BC, CA and AB, respectively of an equilateral triangle ABC. Show that ∆DEF is also an equilateral triangle.
Advertisements
Solution
Given in the question, D, E and F are the mid-points of the sides BC, CA and AB, respectively of an equivalent ∆ABC.
To proof that ∆DEF is an equilateral triangle.
Proof: In ∆ABC, E and F are the mid-points of AC and AB respectively, then EF || BC.
So, EF = `1/2` BC ...(I)
DF || AC, DE || AB
DE = `1/2` AB and FD = `1/2` AC [By mid-point theorem] ...(II)
Now, ∆ABC is an equilateral triangle
AB = BC = CA
`1/2` AB = `1/2` BC = `1/2` CA ...[Dividing by 2 in the above equation]
So, DE = EF = FD ...[From equation (I) and (II)]
Since, all sides of ADEF are equal.
Hence, ∆DEF is an equilateral triangle
Hence proved.
APPEARS IN
RELATED QUESTIONS
ABC is a triang D is a point on AB such that AD = `1/4` AB and E is a point on AC such that AE = `1/4` AC. Prove that DE = `1/4` BC.
ABCD is a quadrilateral in which AD = BC. E, F, G and H are the mid-points of AB, BD, CD and Ac respectively. Prove that EFGH is a rhombus.
In a triangle ABC, AD is a median and E is mid-point of median AD. A line through B and E meets AC at point F.
Prove that: AC = 3AF.
In Δ ABC, AD is the median and DE is parallel to BA, where E is a point in AC. Prove that BE is also a median.
In ΔABC, BE and CF are medians. P is a point on BE produced such that BE = EP and Q is a point on CF produced such that CF = FQ. Prove that: A is the mid-point of PQ.
In ΔABC, D, E and F are the midpoints of AB, BC and AC.
Show that AE and DF bisect each other.
In ΔABC, the medians BE and CD are produced to the points P and Q respectively such that BE = EP and CD = DQ. Prove that: Q A and P are collinear.
In ΔABC, D and E are the midpoints of the sides AB and BC respectively. F is any point on the side AC. Also, EF is parallel to AB. Prove that BFED is a parallelogram.
Remark: Figure is incorrect in Question
P and Q are the mid-points of the opposite sides AB and CD of a parallelogram ABCD. AQ intersects DP at S and BQ intersects CP at R. Show that PRQS is a parallelogram.
Prove that the line joining the mid-points of the diagonals of a trapezium is parallel to the parallel sides of the trapezium.
