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In the given figure, the diameters of two wheels have measures $4 cm$ and $2 cm$ . Determine the lengths of the belts A and BC that pass around the wheels if it is given that belts cross each other at right angles.

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1 cm, 1 cm

2 cm, 2 cm

3 cm, 3 cm

4 cm, 4 cm

answer is C.

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Detailed Solution

It is given that, diameters of the two circles are cm and 2 cm respectively. Thus,

O A = O C = 2 cm

and

O ′ B = O ′ D = 1 cm

.
The tangent drawn at any point of a circle is perpendicular to the radius through the point of contact.
10th Standard CBSE Mathematics - Circles Four Marks Questions | 10th Standard CBSE | Maths question papers | qb365.in

10th Standard CBSE Mathematics - Circles Four Marks Questions | 10th Standard CBSE | Maths question papers | qb365.in

In the triangle OAP and OCP, Since tangent makes 90 degrees with the radius, so,

∠ O A P = ∠ O C P = 90 °

.
By using the SSS property,

Δ O A P ≅ Δ O C P

.
Since belts cross each other at right angles.
Therefore,

∠ A P C = 90 °

.
Also, the sum of angles of a quadrilateral is 360 degrees, so,

∠ A O C = 90 °

Since, opposites sides of the rectangle are equal,

O A = O C, O A = C P

and

O C = A P

.
So,

A P = P C = 2 cm

.
Similarly,

B P = D P = 1 cm

.
So,

A D = A P + D P ⇒ A D = 2 + 1 ⇒ A D = 3 cm

And,

B C = B P + C P ⇒ B C = 1 + 2 ⇒ B C = 3 cm

The required lengths are 3cm and 3cm respectively
Therefore, the correct option is 3.

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