# The radii of two concentric circles are 16 cm and 10 cm. #AB# is a diameter of the bigger circle. #BD# is tangent to the smaller circle touching it at #D#. What is the length of #AD#?

Adopting the origin

here

Solving for

Subtracting the first from the second equation

Finally the sought distance is

or

Explanation:

If

The point

There are many ways to do that. An alternative method is

then

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As per given data the above figure is drawn.

O is the common center of two concentric circles

Let

In

Applying cosine law in

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Since AB is a diameter of the larger circle, its length is twice the radius of the larger circle, which is ( 2 \times 16 ) cm = 32 cm.

Since BD is tangent to the smaller circle touching it at D, and AB is a diameter of the larger circle, triangle ABD is a right triangle with AD as the height, BD as the base, and AB as the hypotenuse.

Using the Pythagorean theorem, we have:

[ AB^2 = AD^2 + BD^2 ]

Substituting the known values, we get:

[ 32^2 = AD^2 + (16 - 10)^2 ]

[ 1024 = AD^2 + 36 ]

[ AD^2 = 1024 - 36 ]

[ AD^2 = 988 ]

[ AD = \sqrt{988} ]

[ AD = 2 \sqrt{247} ]

Therefore, the length of AD is ( 2\sqrt{247} ) cm.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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