10. In the adjoining figure, ABC is a triangle. Through A, B and C lines are drawn parallel to BC, CA and AB respectively, which forms a triangle PQR. Show that 2(AB + BC + CA) = PQ + QR + RP.
Answers
Explanation:
It's just a scale copy of triangle ABC. If you flip the triangle QRP you get a triangle that is a scale copy of the smaller one.
If you multiply the lengthes of each line segment of the smaller triangle, you get the bigger triangle. We can prove this by putting two triangles in it at the bottom. It takes 4 triangles to fill the whole trangle and so therefore we get the triangle QRP.
Answer:
Given : From the figure, it is given that, Through A,B and C lines are drawn parallel to BC , CA and AB respectively.
We have to show that : 2(AB + BC + CA) = PQ + QR + RP
∵ AB ∥ CP and PB ∥CA
Therefore ABPC is a parallelogram
⇒AB = PC and AC = PB (opposite sides of a parallelogram are equal) ---(1)
∵ AB ∥ RC and AR ∥ CB
Therefore, ABCR is a parallelogram.
⇒ AB=CR and CB=AR (opposite sides of a parallelogram are equal) ---- (2)
∵ BC ∥ AQ and AC ∥ BQ
Therefore , ACBQ is a parallelogram
⇒AC = BQ and BC = AQ (opposite sides of a parallelogram are equal) ---(3)
By adding the equations (1), (2) and (3) we get,
AB + AB + AC + AC + BC + BC = PC + CR + BQ + PB + AR + AQ
⇒ 2 AB + 2AC + 2BC = RP + PQ + RQ
∵ ( RP = PC+CR , PQ = PB+BQ , RQ = AR+AQ )
⇒ 2 ( AB + BC + CA ) = PQ + QR + RP
Hence , proved