that they are congruent.
Fig. 6,44
5. D, E and F are respectively the mid-points of sides AB, BC and CA of A ABC. Find the
ratio of the areas of A DEF and A ABC.
Answers
Our aim is to prove that the triangles are similar, in order to find the ratio between the areas of the triangles.
We can say that a side of a triangle is parallel to the line joining mid-points of the other two sides.
Let D and F be the mid-points of AB an AC respectively, in triangle ABC.
Thus DF || BC, then DF || BE too.
By the same process, we can say that E and F are mid-points of both BC and AC respectively.
Then we can also conclude that DF || BC and DF || BE too.
From the above conclusions we also conclude that:
⇒ DF || BE and FE || DB
Thus, we can say that opposite sides of quadrilateral are parallel.
In parallelograms, we conclude that opposite angles are equal.
Hence ∠DFE = ∠ABC.
By the same process, we are able to show that DECF is also a parallelogram.
Opposite sides are equal, in a parallelogram, then:
∠EDF = ∠ACB
From the previous conclusions, we can conclude that:
∠DFE = ∠ABC
∠EDF = ∠ABC
By AA similarity criterion ΔDEF ≅ ΔABC.
The ratio between triangle areas are equal to the square of the ratio of their corresponding side, if two triangles are similar.
Then, we conclude that:
Area of ΔDEF over Area of ΔABC is equal to 1 over 4.
Hence, the ration between its areas is 1/4.
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