A,B,C,D,E,F are 6 consecutive points on the circumference of a circle such that AB=BC=CD=10,DE=EF=FA=22. If the radius of the circle is √n, what is the value of n?

345 dude tell the right answer

268 To find the value of n, which represents the square of the radius of the circle, we need to use the properties of a regular hexagon inscribed in a circle.

Let's start by connecting the consecutive points with lines.

We have:
A --- B --- C --- D --- E --- F --- A

We know that AB = BC = CD = 10, and DE = EF = FA = 22.

Now, draw lines from the center of the circle to each of the six points (A, B, C, D, E, and F) to create radii. Since all radii of a circle are equal, each of these radii will have the same length.

Let O represent the center of the circle. We have:

O
/ \
/ \
/ \
A-------B
\ /
\ /
\ /
C

Since AB = 10, the length of OC will be half of this value, so OC = 5.

Now, focus on quadrilateral ABCO. It is a kite because AO = BO (both are radii) and AC = BC (given in the problem).

Since the diagonals of a kite are perpendicular bisectors of each other, we can find the length of AC by applying the Pythagorean theorem to right triangle ABC:

AC² = AB² - BC²
AC² = 10² - 10²
AC² = 100 - 100
AC² = 0

Since AC = 0, it means that the quadrilateral ABCO is degenerate, meaning that points A, B, C, and O are collinear. In other words, the center of the circle lies on the line segment AB.

Since AB = 10, and AC = 0 (center and A coincide), the radius of the circle is half of the length AB. Therefore, the radius of the circle is 5.

The square of the radius is given by n = 5² = 25.

Therefore, the value of n is 25.