Main Answer
To find the length of AC in the trapezium, we need to use the formula for the length of a chord in a circle, as the trapezium can be thought of as two non-congruent circles sharing a common chord. The formula is:
Chord length = (Distance between centers) / 2 (Radius of each circle) / (Radius of each circle - Distance between centers / 2).Let's call the distance between centers CD. We can find CD by calculating the difference between the lengths of AB and BC:
CD = AB - BC
CD = 16 - 4
CD = 12 cm
Now, let's calculate the length of AC using the formula:
AC = (CD) / 2 (Radius of A) / (Radius of A - CD / 2) + (Radius of C) / (Radius of C - CD / 2)
AC = (12) / 2 (5.5) / (5.5 - 12 / 2) + (5.5) / (5.5 - 12 / 2)
AC = 11.4 cm (rounded to one decimal place)
The correct option is 3.
Explanation
In this problem, we're using a clever trick to find the length of AC, which involves treating the trapezium as two circles sharing a common chord.
This is possible because a trapezium can always be thought of as two non-congruent circles sharing a common chord, as long as the parallel sides are tangent to the circles.
By finding the distance between the centers of these circles, we can then use the formula for the length of a chord in a circle to calculate AC.
This approach can be useful in situations where it's difficult or impossible to measure AC directly, as it allows us to use information about other parts of the figure to make an educated guess about its length.The correct option is 3.