Question

Rohan and Suraj were close friends, One day they were riding horses from Delhi to F Faridabad. The names of their horses were Saku and Fareed respectively. The day was very sunny. On the way, they stopped for resting in a park. They tied their horses to a tree in the park. The length of ropes of Rohans' horse is 14 m and that of the horse of Suraj is 7 m as shown in the figures. Both the friends slept in the park under a green tree for some time. During this period both the horses took 10 rounds along with the tree they were tied.

FIND
(i) (a) The ratio of distance walked in 10 rounds by the horses of Rohan and Suraj
(ii) Prove that the angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle

Answer 1

Answer:

4

1st point only

• pi × r1²÷pi × r2²
• 3.14× 14 ×14 ÷ 3.14 × 7 ×7
• 615.44 ÷ 153.86
• ans 4

Answer 2

Concept

Circles with a same centre are known as concentric circles. An annulus is the area between two concentric circles with differing radii.

By inverting any two circles and selecting the inversion centre as one of the limiting points, any two circles may be made concentric.

Given

Length of rope of Rohan's horse = Radius of outer circle

                                                      = 14m

Length of rope of Suaj's horse = Radius of inner circle

                                                      = 7m

Number of revolutions taken by the horses = 10

Find

1) The ratio of distance walked in 10 rounds by the horses of Rohan and Suraj.

2) To prove that the angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.

Solution

Circumference of a circle is given by the formula;

$Circumference = 2\pi r$

Total distance walked by the horses

$= 10\ .\ circumference\\= 10\ .\ 2\pi r$

Radius of outer circle, $r_{outer} = 14m$

Radius of inner circle, $r_{inner} = 7m$

Ratio of distance walked by the horses

$= \frac{Distance\ travelled\ by\ Rohan's\ horse}{Distance\ travelled\ by\ Suraj's\ horse}\\\\= \frac{10\ .\ 2\pi r_{outer}}{10\ .\ 2\pi r_{inner}}\\= \frac{r_{outer}}{r_{inner}} \\= \frac{14}{7} \\= \frac{2}{1} \\= 2:1$

The ratio of distance walked in 10 rounds by the horses of Rohan and Suraj is 2: 1

Consider an arc PQ of a circle subtending angles POQ at the centre O and PAQ at a point A on the remaining part of the circle.

We have to prove that ∠POQ=2∠PAQ

Construction: Join line AO extended to B.

Now,

∠BOQ = ∠OAQ + ∠AQO       .......(1)

Also, in △OAQ,

OA = OQ                  [Radii of a circle]

Therefore,

∠OAQ = ∠OQA     [Angles opposite to equal sides are equal]

∠BOQ = 2∠OAQ       .......(2)

Similarly,

BOP = 2∠OAP             .......(3)

Adding (2) and (3) we get,

∠BOP + ∠BOQ = 2(∠OAP + ∠OAQ)

∠POQ = 2∠PAQ          

Thus, proved that the angle subtended by an arc at the center (∠POQ ) is double the angle subtended by it at any point on the remaining part of the circle (∠PAQ).