CCSS.Math.Content.HSG-C.A.4
The Standard
(+) Construct a tangent line from a point outside a given circle to the circle.
Common Core State Standards for Mathematics
Teacher's field guide
What This Standard Means
What Students Need to Do
- Students use a compass and straightedge to find the two lines through an outside point that touch a circle once. They explain why each radius to a contact point is perpendicular to its tangent.
What Mastery Looks Like
- Students connect the center to the outside point, bisect that segment, and use the midpoint to locate both tangent points. They draw accurate tangent lines and justify them using right angles.
Common Misconceptions
- Students may draw a line that looks tangent without constructing the contact point. They may place the right angle at the outside point instead of between the radius and tangent.
How to Assess It
- Give students a circle with center O and an outside point P. Ask them to construct both tangents and label the right angles that prove each line is tangent.
Lesson moves
Ways to Teach It
Give pairs compasses, straightedges, and printed circles to construct two tangents from an outside point, keeping every construction mark visible.
Ask students to explain why the circle with diameter from the center to the outside point identifies the tangent points.
Run a construction relay where teams complete and justify tangent constructions from several outside points, earning one point for accuracy and one for reasoning.
Have students design two straight paths from a parking lot to a circular fountain so each path touches the fountain at exactly one point.
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Printable HSG-C.A.4 Worksheet
A ready-to-print activity worksheet aligned to CCSS.Math.Content.HSG-C.A.4, with an answer key for the teacher on its own page. No account needed.
PDF, US Letter, prints cleanly in black and white.
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Where This Standard Sits
Before This Standard
If students are struggling here, check these first.
- CCSS.Math.Content.HSG-CO.C.9
Line and angle theorem reasoning supports the perpendicular right-angle relationships used to justify a tangent construction.
- CCSS.Math.Content.HSG-CO.D.12
Constructing an outside tangent uses formal construction moves like making a midpoint or perpendicular bisector before drawing the tangent lines.
- CCSS.Math.Content.HSG-C.A.2
The tangent construction uses radii and the inscribed angle in a semicircle to locate a right angle at the tangency point.
Keep exploring
Related Standards
- CCSS.Math.Content.HSF-TF.B.7
(+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms...
- CCSS.Math.Content.HSF-TF.A.3
(+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of ...
- CCSS.Math.Content.HSG-GPE.A.3
(+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.
- CCSS.Math.Content.HSF-TF.C.9
(+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
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