Examine the diagram of Circle A. Circle A has a radius of 4, and arc BD has a length of 6.5. Circle C is a different circle with a radius of 6, and it has arc EF. Angle ECF is congruent to angle BAD. What is the length of arc EF? Enter your answer as a number, like this: 42.25.

To determine the length of arc EF in Circle C, we must first understand the relationship between an arc's length, the angle it subtends (the angle formed at the center of the circle by the arc), and the radius of its circle.

Arc length in a circle can be calculated using the formula: \[ \text{Arc length} = \frac{\text{Central angle in degrees}}{360} \times 2\pi \times \text{radius} \]

Given that arc BD in Circle A has a length of 6.5 and a radius of 4, we can use this relationship to find the central angle BAD in Circle A.

Let's assume the measure of angle BAD is θ. The formula for arc length for Circle A with arc BD becomes: \[ 6.5 = \frac{θ}{360} \times 2\pi \times 4 \]

We can solve for θ: \[ θ = \frac{6.5 \times 360}{2\pi \times 4} \]

Calculating that gives us the central angle θ in degrees.

Since angle ECF in Circle C is congruent to angle BAD, they both have the same measure, which is θ.

We can now use the formula above to find the arc length of EF in Circle C, where the radius is given as 6.

So the formula for Circle C with arc EF becomes: \[ \text{Arc EF} = \frac{θ}{360} \times 2\pi \times 6 \]

Since we already have the value of θ calculated, we just need to substitute and compute for the length of arc EF:

\[ \text{Arc EF} = \frac{\text{calculated } θ}{360} \times 2\pi \times 6 \]

You will calculate the actual value using the derived value for θ. Once done, you will have the length of EF as a numerical value.