CCSS.Math.Content.HSG-C.A.3
The Standard
Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
Common Core State Standards for Mathematics
Teacher's field guide
What This Standard Means
What Students Need to Do
- Students use perpendicular bisectors to find a triangle’s circumcenter and draw a circle through all three vertices. They use angle bisectors to find the incenter and draw a circle tangent to all three sides. They also justify angle relationships in quadrilaterals whose vertices lie on one circle.
What Mastery Looks Like
- Students accurately construct both centers with a compass and straightedge, then choose the correct radius for each circle. They prove that opposite angles in a cyclic quadrilateral are supplementary and connect exterior angles to opposite interior angles.
Common Misconceptions
- Students often swap the incenter and circumcenter, or use the wrong bisectors to locate them. They may assume every quadrilateral is cyclic or claim opposite angles are congruent rather than supplementary. Some rely on measurements instead of giving geometric reasons.
How to Assess It
- Give students a printed scalene triangle and ask them to construct both circles. Then ask: In cyclic quadrilateral ABCD, if angle A is 112 degrees, find angle C and justify your answer.
Lesson moves
Ways to Teach It
Give pairs a scalene paper triangle, compass, and straightedge to construct both circles and label every bisector, center, radius, and tangent point.
Ask students to explain why the perpendicular bisectors meet at a point equally distant from all three vertices.
Run a card sort matching circle diagrams, missing angle values, and valid reasons based on intercepted arcs and supplementary angles.
Model a triangular park and decide whether a central sprinkler circle should touch each side or reach each corner.
Free download
Printable HSG-C.A.3 Worksheet
A ready-to-print activity worksheet aligned to CCSS.Math.Content.HSG-C.A.3, with an answer key for the teacher on its own page. No account needed.
PDF, US Letter, prints cleanly in black and white.
Download the worksheetLearning progression
Where This Standard Sits
Before This Standard
If students are struggling here, check these first.
- CCSS.Math.Content.HSG-C.A.2
Inscribed angle relationships support proving cyclic quadrilateral angle properties, though triangle circle constructions mainly use bisectors and perpendicular bisectors.
- CCSS.Math.Content.HSG-CO.D.12
Angle and perpendicular bisector constructions carry directly into locating a triangle’s incenter and circumcenter for the required circles.
- CCSS.Math.Content.HSG-CO.C.10
Triangle angle theorems and proof habits support justifying triangle circle centers and cyclic quadrilateral angle relationships.
Keep exploring
Related Standards
Turn this exact standard into a lesson
Grade, subject, topic, and the complete standard are prefilled. Create one free, no account needed.