HSG.C.A.4: Constructing a Tangent Line from a Point Outside a Circle
In plain English: HSG.C.A.4 is an advanced (+) Common Core geometry standard that asks students to construct, with compass and straightedge, the lines through a point outside a circle that are tangent to the circle. The method draws a circle on the segment from the center to the point as diameter, so each tangent point forms a right angle. It is usually taught in Geometry.
(+) Construct a tangent line from a point outside a given circle to the circle.
Common Core State Standards for Mathematics · Domain: Circles (C) · Cluster: Understand and apply theorems about circles Also written as HSG-C.A.4 or G-C.4 · Official standard
Students learn to construct the tangent lines from a point P outside a circle with center O using only a compass and a straightedge. The construction bisects OP, draws the circle that has OP as a diameter, and joins P to the two points where that circle meets the given circle.
The lesson treats the construction as something to prove, not only to perform. Students justify it with two facts they already know, the right angle inscribed in a semicircle and the perpendicular radius at a point of tangency, and then check their drawings with the tangent length √(OP² - r²).
Learning Objectives
By the end of this lesson, students will be able to:
Construct both tangent lines from a point outside a circle with compass and straightedge
Justify the construction using the inscribed angle in a semicircle and the radius-tangent relationship
Explain why the two tangent segments from an outside point are equal in length
Compute the expected tangent length √(OP² - r²) and use it to check a construction
Describe what happens to the construction when the point is on the circle or inside it
Prior Knowledge Required
Students should already be comfortable with:
Constructing the perpendicular bisector and the midpoint of a segment HSG.CO.D.12
An angle inscribed in a semicircle is a right angle, and a tangent is perpendicular to the radius at the point of tangency HSG.C.A.2
Give each student a sheet with a circle drawn around a marked center O and a point P about 4 cm outside it. Before any instruction, ask them to try the task by eye:
Warm-Up Prompt
"Place your straightedge on P and turn it until the edge just touches the circle. Draw that line. How many different lines like this can you find? How could you prove that your line touches the circle at exactly one point instead of cutting through it?"
Students should find two lines, one on each side of the circle. Collect ideas about proof and steer toward two facts from HSG.C.A.2: a line perpendicular to a radius at its endpoint on the circle is tangent, and an angle inscribed in a semicircle is a right angle. Tell students that "lining it up by eye" is a sketch, not a construction, because nothing guarantees the touch point is exact. The goal of the lesson is a compass-and-straightedge method whose correctness can be proved.
Direct Instruction20 minutes
Model the construction on the board with a large compass while students follow on their own paper. Use Diagram 1 as the finished picture. Name each point as you create it.
Connect the center to the point: draw segment OP.
Find the midpoint of OP: with the compass open to more than half of OP, swing an arc from O and an arc from P on both sides of OP. Draw the line through the two crossings. It is the perpendicular bisector of OP and meets OP at its midpoint M.
Draw the auxiliary circle: put the compass point on M and open it to O. Draw the circle with center M; it also passes through P, so OP is a diameter.
Mark the tangent points: the auxiliary circle crosses circle O at two points. Label them T₁ and T₂.
Draw the tangents: draw line PT₁ and line PT₂. Leave all arcs visible, because they are the record of the construction.
Justification. T₁ lies on the circle with diameter OP, so ∠OT₁P is inscribed in a semicircle and measures 90°. Then line PT₁ is perpendicular to radius OT₁ at T₁, a point on circle O, so PT₁ is tangent to circle O. The same argument works for T₂. Because triangles OT₁P and OT₂P are right triangles with the same hypotenuse OP and equal legs OT₁ = OT₂, they are congruent (HL), so the two tangent segments are equal: PT₁ = PT₂. By the Pythagorean Theorem, each has length √(OP² - r²), which lets students check a construction with a ruler.
Construction with measurements
Circle O has radius 3 cm and P is 5 cm from O (Diagram 1). Where is M, what radius does the auxiliary circle have, and how long should each tangent segment measure?
Equation: OM = 2.5 cm, auxiliary radius 2.5 cm, PT = √(5² - 3²) = 4 cm
Construction on a coordinate grid
Circle O is x² + y² = 25 and P = (13, 0). Carry out the construction with coordinates: find M, the auxiliary circle, and the tangent points.
Equation: M = (6.5, 0), circle (x - 6.5)² + y² = 42.25, T = (25/13, ±60/13), PT = 12
Justifying the right angle
Using the tangent point T = (25/13, 60/13) from the grid example, show that OT is perpendicular to PT.
Equation: OT · PT = (25/13)(25/13 - 13) + (60/13)(60/13) = 0, so ∠OTP = 90°
Both tangent segments
Circle O has radius 8 cm and OP = 17 cm. After the construction, compare PT₁ and PT₂.
Equation: PT₁ = PT₂ = √(17² - 8²) = 15 cm, and OT₁PT₂ is a kite
After the examples, show Diagram 2. If P is on the circle, the tangent is the line through P perpendicular to OP, and the auxiliary circle only touches circle O at P. If P is inside, the circle on diameter OP stays inside circle O, so there are no tangent points and no tangent lines.
Guided Practice15 minutes
Pairs draw circle O with radius 4 cm and mark P 6 cm from O. Partner A performs steps 1-3 while Partner B checks each arc; then they switch for steps 4-5. Before measuring, each pair predicts the tangent length: √(36 - 16) = √20 ≈ 4.47 cm. Pairs measure PT₁ and PT₂ with a centimeter ruler and ∠OT₁P with a protractor, and report how close they came (within 0.1 cm and 2° is good work). Circulate and look for these errors: using the point where OP crosses circle O instead of the midpoint of OP, opening the compass to less than half of OP so the bisector arcs never cross, and drawing the auxiliary circle centered at P.
Independent Practice10-15 minutes
Students complete two constructions on their own. (1) Radius 2.5 cm, P 6.5 cm from O: predict and then measure the tangent length (√(42.25 - 6.25) = 6 cm). (2) Radius 3.5 cm, P 9.1 cm from O: predict and measure (√(82.81 - 12.25) = 8.4 cm). Under each construction, students write a three-sentence justification that names the semicircle and the radius-tangent relationship.
Closure5 minutes
Exit ticket: (1) List the construction steps in order. (2) If OP = 10 cm, to what radius do you open the compass for the auxiliary circle? (5 cm, half of OP.) (3) In one sentence, explain why ∠OTP is a right angle.
Differentiation Strategies
For Struggling Students
Give a step card with a small picture of each stage, and let students check off each arc as they draw it
Start with a large circle (radius 5 cm or more) and a point far from it, so the arcs are easy to see and the intersections are clear
Before the construction, have students draw a right triangle in a semicircle on patty paper to see the right angle for themselves
For Advanced Students
Ask students to prove that the line OP bisects the angle between the two tangents and is the perpendicular bisector of T₁T₂
Ask for the length of the chord T₁T₂ in terms of r and OP, and check it on their construction
Ask students to plan a construction of a line tangent to two separate circles, as a challenge beyond this standard
Assessment Guidance
What to Look For
Look for all construction arcs left visible: the two pairs of bisector arcs, the auxiliary circle and the marked points T₁ and T₂. Ask each student why the line is tangent; a complete answer names both the semicircle (for the right angle) and the radius-tangent relationship (for tangency). Measurement should be used only as a check. A student who adjusts the straightedge until the line "looks" tangent has not met the standard, even if the result is accurate.
02
Classroom Activities
3 Activities
1
Construction Lab: Two Tangents and a Ruler Check
20 minPairs
Students perform the full construction on unlined paper and then use measurement as evidence, not as the method. The ruler only checks the result after the compass and straightedge have done the work.
Procedure
Draw circle O with radius 4 cm. Mark P so that OP = 8.5 cm
Construct the midpoint M of OP with the perpendicular bisector, then the circle with center M through O
Label the two intersection points T₁ and T₂ and draw PT₁ and PT₂
Measure PT₁, PT₂ and ∠OT₁P. The expected values are PT = √(8.5² - 4²) = 7.5 cm and ∠OT₁P = 90°
Discussion Questions
Your measured PT₁ was probably off by a millimeter or two. Does that mean the construction is wrong? What is the difference between the construction and your drawing of it?
Which step guarantees the right angle at T₁?
Why do PT₁ and PT₂ come out equal?
Modification for Distance Learning
Students use a free online compass-and-straightedge tool and share a screenshot that shows every arc. The tool's measurement feature replaces the ruler check.
2
Drag the Point
15 minPairs
In dynamic geometry software, students build the construction once and then drag P to see which parts stay true. This shows that the construction works for every outside point, not only for the one on the worksheet.
Procedure
Draw a circle with center O and radius 6 units and a free point P. Build segment OP, its midpoint M, the circle centered at M through O, the intersection points and the two tangent lines
Drag P until OP = 9, 12 and 15 and record PT each time. Compare with √(OP² - 36): 3√5 ≈ 6.71, 6√3 ≈ 10.39 and 3√21 ≈ 13.75
Drag P slowly toward the circle until OP = 6.5 (PT = 2.5), then onto the circle, then inside it. Record what happens to the auxiliary circle and to the tangent lines
Measure ∠OTP at every position
Discussion Questions
Which measurement never changes as P moves? Why?
What happens to the two tangent lines as P moves far away from the circle?
Why does the construction stop working when P is inside the circle?
Paper Folding Variation
Without software, students draw OP on patty paper and fold P onto O to crease the perpendicular bisector and find M. They finish the auxiliary circle with a compass. HSG.CO.D.12 lists paper folding as a construction method.
3
Proof Card Sort
15 minGroups of 3
Groups receive 8 cards, 4 construction steps and 4 reasons, shuffled. They pair each step with the reason that makes it valid and put the pairs in order. The result is a two-column justification of the construction.
The 8 Cards
Step card: Draw OP and construct its perpendicular bisector to find M
Step card: Draw the circle with center M and radius MO
Step card: Label the points T₁ and T₂ where the two circles meet
Step card: Draw lines PT₁ and PT₂
Reason card: Points on the perpendicular bisector are equidistant from O and P, so M is the midpoint
Reason card: MO = MP, so OP is a diameter of the new circle
Reason card: An angle inscribed in a semicircle is a right angle, so ∠OT₁P = ∠OT₂P = 90°
Reason card: A line perpendicular to a radius at its endpoint on the circle is tangent to the circle
Procedure
Match each step with its reason, then order the four pairs
Write one more line at the end that proves PT₁ = PT₂, and name the congruence criterion you used
Compare with another group and resolve any differences
Challenge Variation
Groups who finish early try to construct a line tangent to two separate circles of different sizes. Tell them it goes beyond this standard, and ask only for a plan and a sketch.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Constructing Both Tangent Lines from P
Drawn to scale with r = 3 cm and OP = 5 cm. The bisector arcs locate M, the midpoint of OP. The dashed circle has diameter OP and meets circle O at T₁ and T₂, where the angles OT₁P and OT₂P are right angles. Each tangent segment is √(5² - 3²) = 4 cm.
Diagram 2: Point Outside, On, or Inside the Circle
From a point outside the circle there are two tangent lines, and the tangent segments are equal. From a point on the circle there is one, perpendicular to the radius. From a point inside there is none: the dashed circle on diameter OP lies inside circle O and never meets it.
04
Homework Assignment
~30 min
HSG.C.A.4 Homework: Tangent Lines from an Outside Point
Directions: Use a compass and straightedge for every construction and leave all arcs visible. A ruler may be used only to measure and check your result. Show every calculation and give lengths to the nearest tenth where they are not whole numbers.
Part 1: Constructing and Justifying (Problems 1-3)
Draw a circle with center O and radius 4.5 cm, and mark a point P that is 7.5 cm from O. Construct both tangent lines from P with a compass and straightedge, leaving every arc visible. Then compute the expected tangent length and compare it with your measurement.
Write a two-column proof that your line PT from Problem 1 is tangent to the circle. Your reasons must include the midpoint of OP, the semicircle, and the relationship between a radius and a tangent line.
Point Q lies on a circle with center O, and point R lies inside it. (a) Describe a construction of the tangent line through Q. (b) Explain, using the auxiliary circle from the outside-point construction, why no tangent line passes through R.
Part 2: Tangent Lengths and Coordinates (Problems 4-6)
Circle O is x² + y² = 144 and P = (0, -20). Carry out the construction with coordinates: give the midpoint M of OP, the equation of the auxiliary circle, and both tangent points. Show that OT ⊥ PT at one of the tangent points, and find the tangent length.
A circular fountain has a radius of 9 feet. A security camera stands 41 feet from the center of the fountain. The two lines of sight from the camera that just graze the edge of the fountain are tangent lines. How far is the camera from each grazing point? Which step of the construction would locate those points on a scale drawing?
Tangent segments PA and PB are drawn from point P to a circle with center O and radius 12 cm, and OP = 37 cm. Find PA and PB, and find the perimeter of quadrilateral OAPB. Explain why OAPB is a kite.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Construction
Both tangents constructed, all arcs and points visible
One tangent or some arcs missing
Drawn by eye or missing
Justification
Names midpoint, semicircle right angle and radius-tangent fact
Reasoning partly complete
No valid reason given
Tangent Lengths
Correct right triangle and lengths, with units
Correct setup with an arithmetic error
Wrong triangle or no work
Special Cases
Point on and inside the circle explained correctly
One case explained
Neither case explained
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Answer each question, then open the explanation to check your reasoning. Your score updates as you go, and Reset quiz clears all answers.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
You are given circle O and a point P outside it. What is the first step of the compass-and-straightedge construction of a tangent line from P?
Answer: B
Every later step depends on segment OP, because the tangent points lie on the circle that has OP as a diameter. Choice A gives a line that is perpendicular to OP at P, which is tangent only when P is on the circle. Choice D is sketching by eye, which does not guarantee a single point of contact.
Question 2 of 20 · Multiple Choice
In the construction, where is the center of the auxiliary circle that locates the tangent points?
Answer: C
The auxiliary circle has diameter OP, so its center is the midpoint of OP, found with the perpendicular bisector. Choice D is a frequent slip: the point where OP crosses the circle is at distance r from O, not OP/2, unless OP happens to equal 2r.
Question 3 of 20 · Multiple Choice
OP = 14 cm. To what radius should you open the compass to draw the auxiliary circle?
Answer: A
The auxiliary circle is centered at the midpoint of OP and passes through O and P, so its radius is 14/2 = 7 cm. Choice B uses the whole segment as the radius, which gives a circle with OP as a radius instead of a diameter. Choice D halves the segment twice.
Question 4 of 20 · Multiple Choice
T is a point where the auxiliary circle meets circle O. Why is ∠OTP a right angle?
Answer: D
T lies on the circle with diameter OP, so ∠OTP intercepts a semicircle and measures 90°. Choice A is false: OT is a radius r and OP is longer. Choices B and C describe relationships that do not exist in the figure.
Question 5 of 20 · Multiple Choice
Once you know ∠OTP = 90°, which fact proves that line PT is tangent to circle O?
Answer: C
PT ⊥ OT and T is on circle O, so PT is tangent at T (the converse of the radius-tangent theorem). Choice A is true but is a consequence of the construction, not the reason PT is tangent.
Question 6 of 20 · Multiple Choice
How many tangent lines to a circle pass through a point outside the circle?
Answer: A
The auxiliary circle crosses circle O at exactly two points, T₁ and T₂, giving two tangent lines (Diagram 2). Choice B describes a point on the circle, and choice C describes a point inside it.
Question 7 of 20 · Multiple Choice
Circle O has radius 7 cm and OP = 25 cm. How long is each tangent segment from P?
Answer: D
△OTP has a right angle at T, with hypotenuse OP: PT = √(25² - 7²) = √576 = 24 cm. Choice A subtracts the lengths instead of their squares, choice B adds them, and choice C adds the squares, treating OP as a leg.
Question 8 of 20 · Multiple Choice
Point P lies on circle O. How do you construct the tangent line through P?
Answer: B
When P is on the circle, OP is a radius, and the tangent at P is the line perpendicular to that radius at P. Choice A is the outside-point method: here the circle with diameter OP touches circle O only at P, so it gives no new point to connect to.
Question 9 of 20 · Multiple Choice
A student tries the construction with P inside circle O. What happens?
Answer: A
Every point of the circle with diameter OP is at most OP from O, and OP is less than the radius, so the two circles never meet. No line through an inside point is tangent. Choice D is false: any segment can be bisected.
Question 10 of 20 · Multiple Choice
Tangent segments from P touch circle O at A and B, with PA = 2x + 5 and PB = 4x - 7. How long is PA?
Answer: C
The construction produces congruent right triangles OAP and OBP (HL), so PA = PB: 2x + 5 = 4x - 7 gives x = 6, and PA = 2(6) + 5 = 17. Choice A stops at x. Choice D substitutes x = 12 after doubling it by mistake.
Question 11 of 20 · Multiple Choice
Circle O has radius r and OP = 2r. What is the angle between the two tangent lines at P?
Answer: D
In right triangle OTP, sin(∠OPT) = r/(2r) = 1/2, so ∠OPT = 30°. The two tangents are symmetric about OP, so the angle between them is 2(30°) = 60°. Choice A gives only half of the angle.
Question 12 of 20 · Multiple Choice
Circle O is x² + y² = 225 and P = (17, 0). Which point is a tangent point of a tangent line from P?
Answer: B
The auxiliary circle is (x - 8.5)² + y² = 72.25. Subtracting it from x² + y² = 225 gives 17x = 225, so x = 225/17 and y = ±120/17. Choice A satisfies x² + y² = 289, so it is not on circle O at all. Choice D is on circle O, but OT and PT are not perpendicular there.
Question 13 of 20 · Multiple Choice
Kai skips the perpendicular bisector and draws a circle centered at P passing through O. He then draws lines from P to the points where this circle meets circle O. Why is his construction wrong?
Answer: C
The right angle at T comes from T being on a circle with diameter OP. On a circle centered at P, OP is a radius, not a diameter, so ∠OTP is not 90° in general and the lines cut through circle O. Choice A ignores the reason the midpoint is needed.
Question 14 of 20 · Multiple Choice
After the construction, quadrilateral OT₁PT₂ has ∠T₁PT₂ = 50°. What is ∠T₁OT₂?
Answer: A
The angles at T₁ and T₂ are 90° each, and the angles of a quadrilateral add to 360°: 360° - 90° - 90° - 50° = 130°. Choice C is the complement of 50°, and choice D subtracts only 50° from 360°.
Question 15 of 20 · Short Answer
Write the construction of the tangent lines from a point P outside circle O as numbered steps, and give the reason that makes each key step work.
Steps: (1) Draw OP. (2) Construct the perpendicular bisector of OP to get its midpoint M. (3) Draw the circle with center M and radius MO; OP is its diameter. (4) Label its intersections with circle O as T₁ and T₂. (5) Draw PT₁ and PT₂. Reasons: ∠OT₁P is inscribed in a semicircle, so it is 90°; PT₁ is perpendicular to radius OT₁ at a point of the circle, so it is tangent. The same holds for T₂.
Question 16 of 20 · Short Answer
Circle O has radius 20 cm and P is 29 cm from O. Find the length of each tangent segment from P.
The radius to the tangent point is perpendicular to the tangent, so △OTP is a right triangle with hypotenuse 29: PT = √(29² - 20²) = √(841 - 400) = √441 = 21 cm for both tangent segments.
Question 17 of 20 · Short Answer
Circle O is x² + y² = 64 and P = (0, 10). Use the construction to find the tangent points and the length of each tangent segment.
M = (0, 5), so the auxiliary circle is x² + (y - 5)² = 25. Subtracting from x² + y² = 64 gives 10y - 25 = 39, so y = 6.4 and x = ±4.8. The tangent points are (4.8, 6.4) and (-4.8, 6.4), and each tangent segment is √(10² - 8²) = 6. Check: OT · PT = 4.8(4.8) + 6.4(6.4 - 10) = 23.04 - 23.04 = 0.
Question 18 of 20 · Short Answer
A tangent segment from P to a circle of radius 15 cm is 20 cm long. How far is P from the center of the circle?
The radius and the tangent segment are the legs of a right triangle with hypotenuse OP: OP = √(15² + 20²) = √625 = 25 cm.
Question 19 of 20 · Short Answer
As P moves closer to circle O, what happens to the auxiliary circle and to the two tangent lines? What happens when P reaches the circle?
As P approaches the circle, the tangent segments get shorter and the two tangent points move toward P, so the tangent lines open up toward a straight angle. When P is on the circle, the circle with diameter OP touches circle O only at P, and the two tangent lines merge into one: the line through P perpendicular to OP.
Question 20 of 20 · Short Answer
Explain why the two tangent segments constructed from P are always equal in length.
Triangles OT₁P and OT₂P both have a right angle (at T₁ and T₂), share the hypotenuse OP, and have legs OT₁ = OT₂ because both are radii. By the HL congruence criterion the triangles are congruent, so PT₁ = PT₂.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does HSG.C.A.4 mean?
HSG.C.A.4 means students can construct, with a compass and straightedge, a line through a point outside a circle that is tangent to the circle. The standard method draws the circle whose diameter joins the center to the outside point; where it meets the given circle are the tangent points. Students should also be able to explain why the lines are tangent.
Is HSG.C.A.4 required for every Geometry student?
Not always: HSG.C.A.4 is marked (+), so Common Core lists it as additional mathematics for students who take advanced courses. Many Geometry courses teach it anyway, often in an honors section, because it uses only ideas already in the circles unit.
Why does the tangent construction use the midpoint of OP?
The midpoint makes OP a diameter of the auxiliary circle. Any point T on that circle forms an angle ∠OTP inscribed in a semicircle, which is a right angle. A right angle at T between the radius OT and the line PT is exactly what makes PT tangent.
Can students just line up a ruler so it touches the circle?
No, not for this standard. Lining up a straightedge by eye produces a sketch whose touch point is only approximate, and nothing in the process can be justified. A construction must be built from steps (arcs and lines) whose results can be proved.
How many tangent lines can be drawn from a point to a circle?
It depends on where the point is: two from a point outside the circle, one from a point on the circle, and none from a point inside. From an outside point, the two tangent segments are equal in length.
What should students know before learning to construct tangent lines?
Students need the basic constructions, especially the perpendicular bisector of a segment (HSG.CO.D.12), and two circle facts from HSG.C.A.2: an angle inscribed in a semicircle is a right angle, and a radius is perpendicular to the tangent at its endpoint. The Pythagorean Theorem is used to check tangent lengths.
What mistakes do students make with this construction?
Common errors are centering the auxiliary circle at P or at the point where OP crosses the circle instead of at the midpoint of OP, opening the compass too little when bisecting OP, drawing only one of the two tangents, and erasing the construction arcs. In calculations, a frequent slip is computing OP - r instead of √(OP² - r²) for the tangent length.
How is the tangent line construction usually assessed?
It is usually assessed by asking students to perform the construction with all arcs visible and then justify it in words or a two-column proof. Teachers also ask related computations, such as the tangent length from the radius and the distance OP, and questions about equal tangent segments.
Can students use geometry software instead of a compass?
Yes. HSG.CO.D.12 lists dynamic geometry software, string and paper folding as construction tools along with compass and straightedge. Software is especially useful for dragging P and seeing that the construction works for every outside point. Many teachers still require one paper construction.
How does constructing tangents connect to later math?
The right triangle formed by the radius, the tangent segment and OP leads to tangent-length and angle problems solved with trigonometry (HSG.SRT.C.8). The idea of a line that touches a curve at one point returns in calculus, where tangent lines to curves describe rates of change.
07
Related Standards
6 standards
These standards connect to HSG.C.A.4: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
HSG.CO.D.12Prerequisite
Make formal constructions: copy, bisect, perpendicular bisector, parallel lines
