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

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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.