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.
Solution:
Given, ABC is an equilateral triangle
D, E and F are the midpoints of the sides BC, CA and AB
We have to show that DEF is also an equilateral triangle
Considering triangle ABC,
E and F are the midpoints of AC and AB
The midpoint theorem states that “The line segment in a triangle joining the midpoint of two sides of the triangle is said to be parallel to its third side and is also half of the length of the third side.”
By Midpoint theorem,
EF || BC
EF = 1/2 BC ------- (1)
DF || AC
DF = 1/2 AC ----------- (2)
DE || AB
DE = 1/2 AB ----------- (3)
Since ABC is an equilateral triangle
The sides AB = BC = AC
Dividing by 2,
1/2 AB = 1/2 BC = 1/2 AC
From (1), (2) and (3)
DE = EF = DF
This implies the sides of triangle DEF are equal.
Therefore, DEF is an equilateral triangle.
✦ Try This: In parallelogram ABCD, two point P and Q are taken on diagonal BD such that DP=BQ. Show that AQ = CP
☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 8
NCERT Exemplar Class 9 Maths Exercise 8.3 Problem 8
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.
Summary:
An equilateral triangle is a triangle in which all the three sides are equal and each angle is equal to 60 degrees. D, E and F are the mid-points of the sides BC, CA and AB, respectively of an equilateral triangle ABC. It is shown that ∆ DEF is also an equilateral triangle
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