P, Q and R are respectively the mid-points of sides BC, AC and AB of an equilateral triangleABC. prove that trianglePQR is also an equilateral triangle.
Answer:
We have equilateral triangle , And have mid points P , Q and R on sides BC , AC and AB respectively , As:
We know In ABC
AB = BC = AC
And
AR = RB = BP = PC = CQ = QA
In BRP , we know
ABC = 60 ( As angle of equilateral ABC )
BPR = BRP ( opposite angle of equal sides as RB = BP )
And
ABC + BPR + BRP = 180
60 + BPR + BRP = 180
2 BPR = 120 ( AS: BPR = BRP )
BPR = 60
Therefore
ABC = BPR = BRP = 60
So BRP is also a equilateral triangle , So
RB = BP = RP
Similarly we can show
PC = CQ = PQ
And,
AR = QA = QR
Therefore
RP = PQ = QR , Thats shows PQR is equilateral triangle . ( Hence proved )
We have equilateral triangle , And have mid points P , Q and R on sides BC , AC and AB respectively , As:
We know In ABC
AB = BC = AC
And
AR = RB = BP = PC = CQ = QA
In BRP , we know
ABC = 60 ( As angle of equilateral ABC )
BPR = BRP ( opposite angle of equal sides as RB = BP )
And
ABC + BPR + BRP = 180
60 + BPR + BRP = 180
2 BPR = 120 ( AS: BPR = BRP )
BPR = 60
Therefore
ABC = BPR = BRP = 60
So BRP is also a equilateral triangle , So
RB = BP = RP
Similarly we can show
PC = CQ = PQ
And,
AR = QA = QR
Therefore
RP = PQ = QR , Thats shows PQR is equilateral triangle . ( Hence proved )