Pregunta
Given circle \( O \) inscribed in triangle \( S T U \) with \( \overline{S Q T}, \overline{T R U} \), and \( \overline{U P S} \) tangent to \( O \) at points \( Q, R \), and \( P \) respectively. Identify a segment that is congruent to \( \overline{Q T} \).
Ask by Gibbs Elliott. in the United States
Mar 11,2025
Solución de inteligencia artificial de Upstudy
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The segment congruent to \(\overline{QT}\) is \(\overline{TR}\).
Solución
1. Notice that in triangle \(STU\) the circle \(O\) is tangent to all three sides.
2. Since the circle is tangent to side \(ST\) at point \(Q\) and to side \(TU\) at point \(R\), vertex \(T\) has two tangent segments: \(\overline{TQ}\) and \(\overline{TR}\).
3. By the tangent-segment theorem, the tangent segments from a common external point to a circle are equal in length.
4. Thus, we have:
\[
TQ = TR
\]
5. This shows that the segment congruent to \(\overline{QT}\) (which is the same as \(\overline{TQ}\)) is \(\overline{TR}\).
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Simplifique esta solución Extra Insights
In triangle \( S T U \) with incircle \( O \), the segments \( \overline{SQ} \), \( \overline{TQ} \), and \( \overline{TP} \) are crucial. The segment \( \overline{SQ} \) is congruent to \( \overline{QT} \) because they both represent the tangential distance from point \( S \) to the point of tangency \( Q \) with respect to side \( ST \). Similarly, \( \overline{QT} \) is also congruent to \( \overline{TR} \), as both segments connect points \( Q \) and \( T \) to the respective points of tangency on the incircle. Thus, in triangle \( S T U \), the segments congruent to \( \overline{QT} \) include \( \overline{SQ} \) and \( \overline{TR} \).
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