In ΔABC,∠A=45∘, ∠B=60∘ and ∠C=75∘. The angles of the triangle formed by joining the mid-points of the sides of this triangle are
Consider the figure below:
D, E and F are the mid-points of the sides BC, AC and AB respectively.
From the converse of mid-point theorem, we can say that
⇒DE∥AB, DF∥AC and EF∥AB
Also, DE=AB2=AF, DF=AC2=AE and EF=AB2=BD
Thus, AFDE,FBDE and EFDC are parallelograms.
In △ABC and △DEF,
∠EDF=∠CAB=45∘ [Opposite angles of parallelogram, AFDE]
∠DEF=∠ABC=60∘ [Opposite angles of parallelogram, FBDE]
and ∠EFD=∠ACB=75∘ [Opposite angles of parallelogram, EFDC]
Hence, the angles of the triangle formed by joining the mid-points of the sides of △ABC are 45∘, 60∘, 75∘.