Question

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 square side length

The perimeter of square ACEG is 24 cm. For a square, side length $s = \frac{\text{Perimeter}}{4}$.
$\displaystyle s = \frac{24}{4} = 6$ cm

Step2: Use tangent segment theorem

Tangents from a single external point to a circle are congruent. Let $AB = AH = x$, $CB = CD = y$, $ED = EF = z$, $GF = GH = w$. Since ACEG is a square, each side is $x + y = y + z = z + w = w + x = 6$ cm. From $x + y = 6$ and $w + x = 6$, we get $y = w$; similarly $x = z$. For the full perimeter, $2(x + y + z + w) = 24$, but since $x + y = 6$, and $CB = y$, we know that in such a tangential square, the tangent segments from each vertex split the side into two equal parts? No, actually, for a square with an incircle (which this is, as it's tangential), the tangents from each vertex to the circle are equal, so each side is split into two segments of equal length. Wait, no—actually, since all sides are equal and tangents from each vertex are congruent, $AB = CB$? No, wait, no: for square ACEG, side AC = 6 cm, which is $AB + BC$. But tangents from A: AB = AH; tangents from C: CB = CD; tangents from E: ED = EF; tangents from G: GF = GH. And since AB + BC = 6, BC + CD = 6, so AB = CD. But since it's a square, the incircle touches each side at its midpoint, so AB = BC = 3 cm. Wait, correction: in a square with an incircle, the point of tangency bisects each side. So each tangent segment from a vertex to the point of tangency is half the side length.
$\displaystyle BC = \frac{6}{2} = 3$ cm

Answer:

B. 3 cm