D, E and F are respectively the mid-points of the sides AB, BC and CA of a triangle ABC. Prove that by joining these mid-points D, E and F, the triangles ABC is divided into four congruent triangles.
Solution:
Given, ABC is a triangle
D, E and F are the midpoints of the sides AB, BC and CA
We have to prove that by joining the midpoints D, E and F, the triangle ABC is divided into four congruent triangles.
Since D is the midpoint of AB
AD = DB = 1/2 AB
Since E is the midpoint of BC
BE = EC = 1/2 BC
Since F is the midpoint of AC
AF = FC = 1/2 AC
By midpoint theorem,
EF || AB
EF = 1/2 AB
So, EF = AD = BD
Also, ED || AC
ED = 1/2 AC
So, ED = AF = FC
Similarly, DF || BC
DF = 1/2 BC
So, DF = BE = CE
Considering triangles ADF and EFD,
AD = EF
AF = DE
Common side = FD
The Side-Side-Side congruence rule states that, if all the three sides of a triangle are equal to the three sides of another triangle then the triangles are congruent.
By SSS criterion, the triangles ADF and EFD are congruent.
Similarly,
The triangles DEF and EDB are congruent.
The triangles DEF and CFE are congruent.
Therefore, the triangle ABC is divided into four congruent triangles.
✦ Try This: Prove that the line segment joining the mid-points of the diagonals of a trapezium is parallel to each of the parallel sides and is equal to half the difference of these sides.
☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 8
NCERT Exemplar Class 9 Maths Exercise 8.4 Problem 16
D, E and F are respectively the mid-points of the sides AB, BC and CA of a triangle ABC. Prove that by joining these mid-points D, E and F, the triangles ABC is divided into four congruent triangles.
Summary:
D, E and F are respectively the mid-points of the sides AB, BC and CA of a triangle ABC. It is proven that by joining these mid-points D, E and F, the triangles ABC is divided into four congruent triangles
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