Inscribed and Circumscribed Circles
Aligned to G-C.A.3 — Common Core State Standards for Mathematics.
What this lesson teaches
The circumscribed circle of a triangle passes through all three vertices; its center is where the perpendicular bisectors of the sides meet. The inscribed circle is tangent to all three sides; its center is where the angle bisectors meet. For a quadrilateral inscribed in a circle, opposite angles are supplementary.
Worked example
A quadrilateral ABCD is inscribed in a circle with angle A = 85°. Find angle C. In a cyclic quadrilateral, opposite angles are supplementary, so angle A + angle C = 180°. Then angle C = 180 - 85 = 95°. The same rule gives angle B + angle D = 180°.
Practice questions
- A cyclic quadrilateral has angle B = 110°. Find angle D.
- Which three lines must you construct to find the center of a triangle's circumscribed circle?
- The inscribed circle of a triangle is tangent to each side. What lines' intersection locates its center?
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
▶ Watch this lessonEn español
Círculos inscritos y circunscritos
El círculo circunscrito de un triángulo pasa por sus tres vértices; su centro está donde se cruzan las mediatrices de los lados. El círculo inscrito es tangente a los tres lados; su centro está donde se cruzan las bisectrices de los ángulos. Para un cuadrilátero inscrito en un círculo, los ángulos opuestos son suplementarios.
Ejemplo: Un cuadrilátero ABCD está inscrito en un círculo con ángulo A = 85°. Halla el ángulo C. En un cuadrilátero cíclico, los ángulos opuestos son suplementarios, así que ángulo A + ángulo C = 180°. Entonces ángulo C = 180 - 85 = 95°. La misma regla da ángulo B + ángulo D = 180°.
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