Advertisements
Advertisements
प्रश्न
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
उत्तर
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
संबंधित प्रश्न
Show that the line segments joining the mid-points of the opposite sides of a quadrilateral
bisect each other.
A parallelogram ABCD has P the mid-point of Dc and Q a point of Ac such that
CQ = `[1]/[4]`AC. PQ produced meets BC at R.
Prove that
(i)R is the midpoint of BC
(ii) PR = `[1]/[2]` DB
In parallelogram ABCD, E and F are mid-points of the sides AB and CD respectively. The line segments AF and BF meet the line segments ED and EC at points G and H respectively.
Prove that:
(i) Triangles HEB and FHC are congruent;
(ii) GEHF is a parallelogram.
In ΔABC, AB = 12 cm and AC = 9 cm. If M is the mid-point of AB and a straight line through M parallel to AC cuts BC in N, what is the length of MN?
The diagonals of a quadrilateral intersect each other at right angle. Prove that the figure obtained by joining the mid-points of the adjacent sides of the quadrilateral is a rectangle.
In ΔABC, P is the mid-point of BC. A line through P and parallel to CA meets AB at point Q, and a line through Q and parallel to BC meets median AP at point R. Prove that: BC = 4QR
In ΔABC, D, E and F are the midpoints of AB, BC and AC.
If AE and DF intersect at G, and M and N are the midpoints of GB and GC respectively, prove that DMNF is a parallelogram.
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 ∆ABC, AB = 5 cm, BC = 8 cm and CA = 7 cm. If D and E are respectively the mid-points of AB and BC, determine the length of DE.
P, Q, R and S are respectively the mid-points of sides AB, BC, CD and DA of quadrilateral ABCD in which AC = BD and AC ⊥ BD. Prove that PQRS is a square.
