The small circle and the large circle have the same center O. Chord AB is tangent to the smaller circle at C. Prove that the area between the large and small circle is the same as the area of the circle that has AB as a diameter

If the two circles have radii r and R, then the area between the circles is

π(R^2-r^2)

If a circle has AB as a diameter, then BC is its radius, since
AB = AC + BC

BC^2 + r^2 = R^2
so,

BC^2 = R^2-r^2

That means the area of the circle with BC as its radius (AB as its diameter) is

π BC^2 = π(R^2-r^2)

To prove that the area between the large circle and the small circle is the same as the area of the circle that has chord AB as a diameter, we can follow a step-by-step proof:

Step 1: Draw a diagram:
Start by drawing a large circle and a smaller circle with the same center O. Label the points where chord AB intersects the large circle as D and E. Label the tangent point on the smaller circle as C.

Step 2: Identify the radii and diameters:
Let r be the radius of the smaller circle and R be the radius of the larger circle. Since both circles have the same center, OC is equal to OD = OE = R. The line AC is perpendicular to AB, so AC is also equal to r.

Step 3: Find the area between the circles:
The area between the circles is the difference between the areas of the two circles. The area of the larger circle is πR^2, and the area of the smaller circle is πr^2.

Step 4: Find the area of the circle with AB as a diameter:
Since AB is a diameter, the radius of this circle is AB/2, which is equal to OC. Therefore, the area of this circle is π(OC)^2 = πr^2.

Step 5: Show that the areas are equal:
To prove that the areas are equal, we need to show that πR^2 - πr^2 is equal to πr^2.

Since we know that OC = OD = OE = R, we can find AC using the Pythagorean theorem:
AC^2 = OC^2 - OA^2
r^2 = R^2 - (R - r)^2
r^2 = R^2 - (R^2 - 2Rr + r^2)
r^2 = 2Rr - R^2

Multiply both sides by π:
πr^2 = 2πRr - πR^2

Now substitute this into the equation for the area between the circles:
πR^2 - πr^2 = πR^2 - (2πRr - πR^2) = 2πRr - πR^2 = πr^2

Thus, the areas are equal, which confirms the proof.