Henry needs to give an informal proof of the formula for the circumference of a circle.
● He first constructs a circle, with center O, and labels a point on the circle as P.
● He draws a radius from O to P.
● He then uses point P as the center to construct a new circle.
● He draws two line segments, each formed by joining point O with the points of intersection of the two circles.
Which of these is a plausible next step in Henry's proof process?
A. Construct another circle with a doubled radius.
B. Construct another circle with a doubled radius.
C. Construct a rectangle that circumscribes the original circle.
D. Construct a rectangle that circumscribes the original circle.
Construct an octagon that circumscribes the original circle.
Construct an octagon that circumscribes the original circle.
Construct a hexagon inscribed in the original circle.
c
C. Construct a rectangle that circumscribes the original circle.
The plausible next step in Henry's proof process would be to construct a hexagon inscribed in the original circle.
To understand why, let's analyze the steps Henry has taken so far. He started by constructing a circle with center O and labeled a point on the circle as P. Then, he drew a radius from O to P, and used point P as the center to construct a new circle. Finally, he drew two line segments, each formed by joining point O with the points of intersection of the two circles.
By connecting the points of intersection between the original circle and the new circle, Henry has created a regular hexagon inscribed in the original circle. A regular hexagon is a polygon with six equal sides and six equal angles. In this case, the sides of the hexagon are equal to the radius of the circle.
This step is important because it allows Henry to divide the circle into six equal parts, each representing an angle of 60 degrees. By doing so, Henry can then use the knowledge that the sum of the interior angles of a hexagon is 720 degrees, to determine that each angle in the original circle is 60 degrees.
The formula for circumference of a circle, C, is derived from the relationship between the length of the circumference and the measure of the angle formed at the center of the circle. In this case, since each angle in the original circle is 60 degrees, Henry can use the formula C = 2πr, where r is the radius of the circle, to find the circumference. By multiplying the radius by 2π, Henry will obtain the circumference of the circle.
Therefore, based on the steps Henry has already taken and the objective of proving the formula for the circumference of a circle, the plausible next step would be to construct a hexagon inscribed in the original circle.