Pregunta
upstudy study bank question image url

Which of the following statements is true regarding the relationship between circles and triangles? A. There is only one unique circle that can be circumscribed about a triangle. B. There are many circles that can be inscribed in a given triangle. C. There are many circles that can be circumscribed about a triangle. D. There are many triangles that can be inscribed in a given circle.

Ask by Lee Burgess. in the United States
Mar 12,2025

Solución de inteligencia artificial de Upstudy

Respuesta verificada por el tutor

Responder

There is only one unique circle that can be circumscribed about a triangle.

Solución

**Step 1. Analyze Statement A** Given any triangle, a circle that passes through all three vertices is called the circumcircle. One of the fundamental theorems in geometry shows that the perpendicular bisectors of the sides of a triangle are concurrent. Their point of concurrency, called the circumcenter, is the center of the unique circumcircle. Thus, there is exactly one circle that can be circumscribed about a given triangle. In mathematical terms, if a triangle has vertices \(A\), \(B\), and \(C\), then the perpendicular bisectors of segments \(\overline{AB}\), \(\overline{AC}\), and \(\overline{BC}\) intersect at one unique point \(O\). The circle with center \(O\) and radius \(OA\) (or equivalently \(OB\) or \(OC\)) is the unique circumcircle. **Step 2. Analyze the Other Options** - **Statement B:** “There are many circles that can be inscribed in a given triangle.” In any triangle, the circle tangent to all three sides is called the incircle, and it has a unique location. The angle bisectors of the triangle are concurrent at the incenter, which is the center of the incircle. Thus, there is only one incircle, not many. - **Statement C:** “There are many circles that can be circumscribed about a triangle.” As described in Step 1, the circumcircle is unique. There is not more than one circle that passes through all three vertices of a triangle. - **Statement D:** “There are many triangles that can be inscribed in a given circle.” It is true that if you are given a circle, you can form infinitely many triangles by selecting any three non-collinear points on the circle. While this statement is correct, the well-known relationship between circles and triangles usually refers to the unique circumcircle associated with a given triangle. **Step 3. Determine the Best Answer** Both Statements A and D are true. However, the special and often emphasized geometric relationship is that every triangle has one and only one unique circumscribed circle (the circumcircle). This is a fundamental fact in Euclidean geometry. Thus, the statement that best captures this unique relationship is: \( \textbf{A. There is only one unique circle that can be circumscribed about a triangle.} \)

Respondido por UpStudy AI y revisado por un tutor profesional

error msg
Explicar
Simplifique esta solución

Mind Expander

The first additional information: The unique circle that can be circumscribed about a triangle is known as the circumcircle, which passes through all three vertices of the triangle. The center of this circle is called the circumcenter, and it can be determined by the intersection of the perpendicular bisectors of the triangle's sides. This fascinating geometric relationship guarantees that no matter how you tweak the triangle, the circumcircle will always exist! The second additional information: While it's true that a triangle can have only one circumcircle, the opposite is true for inscribed circles. A triangle has exactly one incircle that touches each of its sides at precisely one point, known as the incenter. This circle perfectly fits inside the triangle, illustrating the wonderful balance and harmony of geometric figures! So, when you're exploring triangles, remember their unique partnership with circles!

preguntas relacionadas

Latest Geometry Questions

¡Prueba Premium ahora!
¡Prueba Premium y hazle a Thoth AI preguntas de matemáticas ilimitadas ahora!
Quizas mas tarde Hazte Premium
Estudiar puede ser una verdadera lucha
¿Por qué no estudiarlo en UpStudy?
Seleccione su plan a continuación
Prima

Puedes disfrutar

Empieza ahora
  • Explicaciones paso a paso
  • Tutores expertos en vivo 24/7
  • Número ilimitado de preguntas
  • Sin interrupciones
  • Acceso completo a Respuesta y Solución
  • Acceso completo al chat de PDF, al chat de UpStudy y al chat de navegación
Básico

Totalmente gratis pero limitado

  • Solución limitada
Bienvenido a ¡Estudia ahora!
Inicie sesión para continuar con el recorrido de Thoth AI Chat
Continuar con correo electrónico
O continuar con
Al hacer clic en "Iniciar sesión", acepta nuestros términos y condiciones. Términos de Uso & Política de privacidad