Question

cr sem b geometry
special segments
line segment st is congruent to which line segment?
\overline{tk}
\overline{sk}
\overline{ku}
\overline{ut}

Explanation:

Step1: Recall Tangent Properties

Tangents from a common external point to a circle are congruent. Point \( T \) is external to the circle with center \( K \). \( \overline{ST} \) and \( \overline{UT} \) are tangents from \( T \) to the circle. Wait, no, wait—wait, looking at the diagram, \( SK \) and \( KU \) are radii, so \( SK = KU \). \( TK \) is a common side. Wait, actually, \( \triangle STK \) and \( \triangle UTK \) are congruent (SSS: \( SK = KU \), \( TK = TK \), \( ST = UT \)? Wait, no, the question is which segment is congruent to \( ST \). Wait, \( T \) is outside the circle, \( ST \) and \( UT \) are tangents, so \( ST = UT \). Wait, the options are \( \overline{TK} \), \( \overline{SK} \), \( \overline{KU} \), \( \overline{UT} \). So \( ST \cong UT \) because tangents from a common external point to a circle are congruent. So the correct segment is \( \overline{UT} \).

Step2: Identify Congruent Segment

From the tangent - radius property and congruent triangles (or tangent segment property), tangents from a single external point to a circle have equal length. So \( ST = UT \), meaning \( \overline{ST} \cong \overline{UT} \).

Answer:

\(\overline{UT}\) (the fourth option, labeled as \(\overline{UT}\))