In above figure we see points P and Q, and two paths connecting them. Class 9

In above figure we see points P and Q, and two paths connecting them. Class 9

Question 1.

In above figure we see points P and Q, and two paths connecting them. Class 9

The first path is made up of the semicircle a.

The other path is made up of three semicircles (b, c and d)

Which path is longer: Choose one

i. Path a is longer

ii. Path b + c + d is longer

iii. The two paths have equal length?

Answer:

The length of a semicircle is given by $\frac{1}{2}$ × 2πr = πr

Let the radii of semicircles a, b, c and d be a', b', c', d' respectively.

Length of path a = πa' ... (i)

Length of second path

(b + c + d) = πb'+ πc' + πd' = π(b' + c' + d') ...(ii)

Since both paths join the same points P and Q, the total distance

PQ = a = b + c + d

a = b + c + d ...(iii)

From (i), (ii) and (iii), we get

πa' = π(b’ + c' + d')

Hence, both paths have equal length.