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 6 cm 8 cm

Explanation:

Step1: Find the side length of the square

The perimeter formula of a square is \(P = 4s\) (where \(P\) is the perimeter and \(s\) is the side - length). Given \(P=24\) cm.

$$s=\frac{P}{4}=\frac{24}{4}=6$$

So, \(AC = 6\) cm.

Step2: Use the property of tangents

Since all tangents to the circle from an external point are congruent. Let the point of tangency divide the sides of the square. Let the length of \(BC=x\).
We know that \(AC\) is the side of the square. If we assume the equal - tangent lengths, and since the square is formed by congruent tangents, and the side of the square \(AC\) is composed of two equal - length tangent segments (from the property of tangents to a circle from an external point). If \(AC = 6\) cm and \(AC=2BC\) (because of the equal - tangent property in a square circumscribing a circle)

$$BC=\frac{AC}{2}$$

$$BC = 3$$

Answer:

\(3\) cm