Question

Two poles of heights 5m and 3m are erected upright on the ground and ropes are strected from the top of each to the foot of the other. At what height above the ground do the ropes cross each other?​

Answer 1

Answer:

Solution:-

Let the height of the two poles be 'a' and 'b' meters respectively and let them be 'p' meters apart.

Height of the pole AB = a meter

Height of the pole CD = b meter

Distance between the poles = p meters

Let the point of intersection of lines joining the top of the poles be 'E' and its height be 'h' meters. Suppose BF = x meter.

Draw EF ⊥ BC

In Δ ABC and Δ EFC,

∠ ACB = ∠ ECF      (Common)

∠ ABC = ∠ EFC      (90°)

∴ Δ ABC ~ Δ EFC   (AA similarity)

⇒ AB/EF = AC/EC = BC/FC  (Corresponding sides are proportional)

⇒ a/h = p/(p - x)

⇒ ap - ax = ph  .....(1)

Similarly,  Δ DCB ~ Δ EFB

⇒ b/h = p/x

⇒ x = ph/b  ......(2)

From (1) and (2), we get

ap - aph/b = ph

⇒ h(1 + a/b) = a

⇒ h{(a + b)/b} = a

⇒ h = ab/(a + b)

Or, h = pq/(p + q)

Hence proved.

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