Question

all tangents to the circle are congruent and form a square. the perimeter of square aceg is 24 cm. what is the length of line segment bc? 2 cm 3 cm 8 cm 6 cm

Explanation:

Step1: Find side - length of square ACEG

Since the perimeter of square ACEG is 24 cm, and the perimeter of a square is $4s$ (where $s$ is the side - length), we have $4s = 24$. Solving for $s$, we get $s=\frac{24}{4}=6$ cm.

Step2: Use property of tangents from an external point

Let the center of the circle be $O$. Tangents drawn from an external point to a circle are equal. Let the points of tangency be as shown. Let the side - length of the square ACEG be $a = 6$ cm. Let the lengths of the tangents be $x,y,z,w$ such that $AE=EG = GC=CA = 6$ cm.
We know that if we consider the tangents from the vertices of the square to the circle, we can use the fact that the sum of opposite sides of a quadrilateral circumscribing a circle are equal. In our case, since the outer - most figure is a square and the inner figure is a circle, we can also note that the lengths of the segments between the points of tangency on the sides of the square are equal.
Let's assume that the segments of the sides of the square formed by the points of tangency with the circle divide the sides of the square into equal - length sub - segments. Since the square ACEG circumscribes the circle and all tangents to the circle are congruent, we can see that the length of BC is half of the side - length of the square ACEG. So $BC = 3$ cm.

Answer:

3 cm