HSG.CO.D.12: Formal Geometric Constructions with Compass, String, Folding and Software
In plain English: HSG.CO.D.12 is the Common Core geometry standard that asks students to make formal geometric constructions with tools such as compass and straightedge, string, reflective devices, paper folding and geometry software. Students copy segments and angles, bisect segments and angles, and construct perpendicular lines and parallel lines. It is usually taught in high school Geometry.
Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
Common Core State Standards for Mathematics · Domain: Congruence (CO) · Cluster: Make geometric constructions Also written as HSG-CO.D.12 or G-CO.12 · Official standard
Students make the six basic constructions named in HSG.CO.D.12: copying a segment, copying an angle, bisecting a segment, bisecting an angle, constructing perpendicular lines (including the perpendicular bisector of a segment), and constructing a line parallel to a given line through a point not on it. A formal construction uses tools that copy distances or reflect figures, never a ruler scale or protractor reading, so the result is exact rather than measured.
Students work with several tools, as the standard asks: compass and straightedge, string and pushpin, paper folding, a reflective device and dynamic geometry software. For each construction they also say why it works, using the perpendicular bisector theorem, SSS congruence or the converse of the corresponding angles theorem. Rulers and protractors are used only afterwards, to check.
Learning Objectives
By the end of this lesson, students will be able to:
Copy a segment and copy an angle with compass and straightedge, and explain why the copies are congruent
Bisect a segment and bisect an angle with compass and straightedge and with paper folding
Construct the perpendicular bisector of a segment and the perpendicular to a line through a point on or off the line
Construct a line parallel to a given line through a point not on the line, and justify it with the converse of the corresponding angles theorem
Carry out constructions with string, a reflective device and dynamic geometry software, and compare the methods
Prior Knowledge Required
Students should already be comfortable with:
Precise definitions of circle, angle, perpendicular lines, parallel lines and segment HSG.CO.A.1
The perpendicular bisector theorem and its converse, and the corresponding angles theorem HSG.CO.C.9
Hand each student a strip of paper with a pencil segment drawn on it, and take away the rulers.
Warm-Up Prompt
"Without a ruler or any measuring, find the exact midpoint of the segment on your strip. Then explain why your point is exactly in the middle and not just close."
Most students fold the strip so that the endpoints meet. Ask what the crease is (the perpendicular bisector: folding is a reflection that takes one endpoint to the other). Explain that a construction is a procedure that is exact because of a reason, not because of careful measuring. Today students learn six of them and several tools for each.
Direct Instruction20 minutes
The tools. A straightedge draws lines but does not measure. A compass draws circles and carries a distance from one place to another; a string held by a pushpin does the same job. Paper folding makes reflections: the crease is the line of reflection. A reflective device (such as a Mira) shows a reflection, so its edge can be placed on a line of reflection. Dynamic geometry software has circle, line and intersection tools that act like compass and straightedge. Model each construction under a document camera.
Copy a segment AB: draw a ray from C, set the compass to AB, and draw an arc centered at C that crosses the ray at D. Then CD = AB.
Copy an angle ∠A: draw a ray from a new vertex A'. With any opening, draw an arc centered at A crossing both sides at B and C, and the same arc centered at A' crossing the ray at B'. Set the compass to BC and draw an arc centered at B' crossing the first arc at C'. Draw ray A'C'.
Bisect a segment AB (perpendicular bisector): with an opening greater than half of AB, draw arcs centered at A and at B on both sides of AB. The line through the two crossing points is the perpendicular bisector, and it crosses AB at the midpoint. By folding: fold so that A lands on B.
Bisect an angle: draw an arc centered at the vertex V crossing the sides at X and Y. With one opening, draw arcs centered at X and at Y that cross at Z. Ray VZ is the bisector. By folding: fold so that one side lies on the other.
Perpendicular through a point K: draw an arc centered at K crossing the line at G and H, then construct the perpendicular bisector of GH, which passes through K. This works whether K is on the line or not. By folding: fold the line onto itself so that the crease passes through K.
Parallel through a point P not on line ℓ: draw a transversal through P meeting ℓ at Q, then copy the angle the transversal makes with ℓ at Q to P in the corresponding position. Another method: construct the perpendicular to ℓ through P, then the perpendicular to that line at P.
Copy a segment
Segment AB is 6.4 cm long. Copy it onto a ray that starts at C, using only a compass and a straightedge, and explain why the copy is exact.
Equation: D is on the circle centered at C with radius AB, so CD = AB = 6.4 cm. No ruler reading is needed
Bisect a segment
AB = 8 cm. Arcs with a 5 cm opening are drawn from A and from B and cross at P and Q (Diagram 1). How far are P and Q from the midpoint M, and why is PQ the perpendicular bisector?
Equation: PM = √(5² - 4²) = 3 cm. PA = PB = 5 and QA = QB = 5, so by the converse of the perpendicular bisector theorem P and Q lie on it
Copy an angle
Copy ∠DEF, which measures 48°, to a ray from K. Explain why the copy also measures 48°.
Equation: The two arcs have equal radii and the chord is copied, so the two triangles are congruent by SSS and the copied angle is 48°
Bisect an angle
Bisect a 70° angle with vertex V (Diagram 1). What does each half measure, and why is the construction exact?
Equation: VX = VY and XZ = YZ, so △VXZ ≅ △VYZ by SSS and ∠XVZ ≅ ∠YVZ: each half is 70° ÷ 2 = 35°
Parallel through a point
A transversal through P meets line ℓ at Q, making a 60° angle with ℓ. Construct the line k through P parallel to ℓ (Diagram 2).
Equation: Copy the 60° angle at P as a corresponding angle. Corresponding angles are congruent, so k ∥ ℓ by the converse of the corresponding angles theorem
After each example, ask the class: "What makes this exact?" Collect the three reasons that support every construction on the page: equal radii of one circle, the converse of the perpendicular bisector theorem, and SSS congruence (plus the corresponding angles converse for parallels).
Guided Practice15 minutes
Pairs construct the perpendicular to a line m through a point K that is 4 cm from m, first with compass and straightedge and then by folding. Walk through the compass version together:
Set the compass to 5 cm and draw an arc centered at K that crosses m at G and H. Predict GH before drawing: each crossing is √(5² - 4²) = 3 cm from the foot of the perpendicular, so GH = 6 cm.
Open the compass wider than 3 cm and draw arcs centered at G and at H on the far side of m. Label their crossing L.
Draw line KL. Check with a protractor that it meets m at 90°, and check with a ruler that GH is about 6 cm.
Fold a second copy of the figure so that m lands on itself and the crease passes through K. Compare the crease with line KL.
Ask pairs to write one sentence explaining why KL is perpendicular to m (K and L are each equidistant from G and H). Watch for students who change the compass opening between the arcs at G and at H: the two arcs must have the same radius.
Independent Practice15 minutes
Students construct on their own and then check with a ruler or protractor. (1) Copy a 5.5 cm segment, then bisect the copy. (Each half should measure 2.75 cm.) (2) Draw a 110° angle, bisect it, then bisect one of the halves. (The halves are 55°, and the quarter is 27.5°.) (3) By paper folding only, construct a line through a point P parallel to a line ℓ: fold the perpendicular to ℓ through P, then fold the perpendicular to that crease through P. Explain why the second crease is parallel to ℓ. (Both ℓ and the second crease are perpendicular to the first crease, so corresponding angles are both 90° and the lines are parallel.)
Closure5-10 minutes
Exit ticket: (1) Name the two constructions in this lesson whose justifications use SSS (copying an angle and bisecting an angle). (2) Name one tool other than a compass that can bisect a segment, and explain how. (3) Why must the compass opening be more than half of AB when you bisect segment AB?
Differentiation Strategies
For Struggling Students
Give step cards with a picture of each step, and let students check off steps as they go
Start with paper folding for bisectors and perpendiculars, where the reflection is visible, before using the compass
Provide compasses that lock their opening, so the radius does not slip between arcs
For Advanced Students
Construct a segment whose length is the sum or difference of two given segments, then one whose length is 1.5 times a given segment
Construct a 45° angle and a 22.5° angle from a line, using only perpendiculars and angle bisectors
Use dynamic geometry software to construct a parallel line, then drag the original points and explain why the construction keeps the lines parallel while a line drawn by eye does not
Assessment Guidance
What to Look For
Check that construction marks (arcs) are left visible, since they are the evidence that the figure was constructed, not drawn by eye. The compass opening must stay the same where the construction needs equal radii. Strong explanations name the reason: equal radii of one circle, the converse of the perpendicular bisector theorem, SSS, or the converse of the corresponding angles theorem. Students should also be able to do at least one construction with a second tool.
02
Classroom Activities
3 Activities
1
Five Tools, Five Stations
20 minGroups of 3-4
Groups rotate through 5 stations, 4 minutes each. Each station has one tool and one construction, so every group uses all the tools the standard lists.
The 5 Stations
Station 1, compass and straightedge: copy a given angle onto a ray
Station 2, string and pushpin: pin the string to cardboard and use it as a compass to copy a segment and then bisect it
Station 3, paper folding: fold the bisector of an angle drawn on patty paper, and fold a perpendicular to a line through a marked point
Station 4, reflective device: place a Mira so that the image of A falls exactly on B, and draw along its edge to get the perpendicular bisector of AB
Station 5, dynamic geometry software: construct a line parallel to a given line through a point by copying a corresponding angle with the circle tool
Procedure
At each station, one student constructs, one checks with a ruler or protractor, and one records the reason the construction works
Rotate roles at every station
Groups leave their work at each station so the next group can compare
Discussion Questions
Which tool acts as a compass, and which tools act as reflections?
Which tool was the most precise for you, and why?
2
Construct and Verify
20 minPairs
Each pair gets 6 task cards, one for each construction in the standard. One partner constructs with compass and straightedge while the other waits. Then they swap papers, and the checker measures and names the reason the construction works. Partners switch roles after each card.
The 6 Task Cards
Card 1: copy a 7.2 cm segment (check: 7.2 cm)
Card 2: copy a 38° angle (check: 38°)
Card 3: bisect a 9 cm segment (check: two 4.5 cm halves and a 90° angle)
Card 4: bisect a 126° angle (check: two 63° angles)
Card 5: construct the perpendicular to a line through a point 5 cm away from it (check: 90°)
Card 6: construct the line through a point parallel to a given line (check: corresponding angles congruent)
Procedure
The checker accepts a construction if it is within 1 mm or 1° and all arcs are visible
The checker writes the reason on the card: equal radii, the perpendicular bisector converse, SSS or the corresponding angles converse
If a construction is off, the pair finds the step that went wrong
Modification for Distance Learning
Pairs do the cards in shared dynamic geometry software, using only the circle, line, segment and intersection tools, and then use the measure tools to check.
3
Why Does It Work?
15 minGroups of 3
Groups match each construction with the theorem that makes it exact, then write a short paragraph justification for two of them.
Match
Copy a segment: every point of a circle is the same distance from its center
Copy an angle: SSS, then corresponding parts of congruent triangles
Bisect a segment: the converse of the perpendicular bisector theorem
Bisect an angle: SSS, then corresponding parts of congruent triangles
Perpendicular through a point: the converse of the perpendicular bisector theorem (applied to GH)
Parallel through a point: the converse of the corresponding angles theorem
Challenge Variation
Give students the steps of the parallel construction by two perpendiculars. They write a justification that uses only the perpendicular construction and the fact that two lines perpendicular to the same line are parallel.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Bisecting a Segment and Bisecting an Angle
Left, to scale: arcs of radius 5 cm centered at A and B cross at P and Q, 3 cm from the midpoint M, and line PQ is the perpendicular bisector of the 8 cm segment AB. Right: an arc centered at V crosses the sides of a 70° angle at X and Y, equal arcs from X and Y cross at Z, and ray VZ bisects the angle into two 35° angles.
Diagram 2: Constructing Parallel and Perpendicular Lines
Left: the 60° angle ∠SQR is copied at P with the same compass openings, so line k through P is parallel to ℓ. Right: an arc centered at K crosses m at G and H, and equal arcs from G and H cross at L; K and L are both equidistant from G and H, so line KL is perpendicular to m.
Directions: Use a compass and a straightedge unless a problem names another tool. Leave all construction marks visible. Use a ruler or protractor only to check your result, and give the reason each construction works.
Part 1: Segments (Problems 1-3)
Draw a 9 cm segment AB and construct its perpendicular bisector with a 6 cm compass opening. (a) Predict the distance from each arc crossing to the midpoint, then measure it. (b) Explain why a 4 cm opening would not work.
Draw a segment of length a = 4 cm and a segment of length b = 2.5 cm. Using only compass and straightedge, construct a segment of length 2a - b. Describe your steps and give the length of the result.
Describe how to bisect a segment drawn on cardboard using only a string and a pushpin (no compass), and explain why the string can replace the compass. Then describe a second method that uses paper folding.
Part 2: Angles, Perpendiculars and Parallels (Problems 4-6)
Draw a 128° angle. (a) Copy it onto a new ray. (b) Bisect the copy, then bisect one of the halves. Give the measure of each angle you construct. (c) Explain with SSS why the copy in part (a) is congruent to the original.
Draw a line ℓ and a point K that is 5 cm from ℓ. Construct the perpendicular to ℓ through K using a 7 cm compass opening for the first arc. (a) Predict the distance between the two points where the arc crosses ℓ. (b) Then construct the perpendicular to ℓ through a point that lies on ℓ.
Draw a line ℓ and a point P not on ℓ. Construct the line through P parallel to ℓ in two ways: (a) by copying a corresponding angle with compass and straightedge, and (b) by constructing two perpendiculars, either by paper folding or in dynamic geometry software. Justify each method with a theorem.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Accuracy
Within 1 mm and 1° of the target
Within 3 mm or 3°
Not close, or drawn by eye
Construction Marks
All arcs and creases visible
Some marks erased
No marks
Justification
Correct theorem named for each construction
Reason given but incomplete
No reason
Variety of Tools
Second tool used correctly where asked
Second tool described but not carried out
Only one tool
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Choose an answer for each multiple-choice question to see whether it is right and why. For the short answers, write your steps and reasons on paper (and try the construction) before opening the model answer. 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
To copy segment AB onto a ray that starts at point C, how should the compass be set before drawing the arc centered at C?
Answer: C
The arc centered at C must have radius AB, so every point on it, including where it crosses the ray, is AB away from C. Choice A gives a segment of the wrong length. Choice B gives half the segment. Choice D copies the wrong distance.
Question 2 of 20 · Multiple Choice
Segment AB is 10 cm long. Which compass opening can be used for both arcs to construct its perpendicular bisector?
Answer: D
The arcs centered at A and B must cross at two points, so the opening must be more than half of AB, which is 5 cm. Only 6 cm works. With 3 cm or 4 cm the arcs do not meet. With exactly 5 cm they touch only at the midpoint, which gives one point, and a line needs two.
Question 3 of 20 · Multiple Choice
A 24 cm segment is bisected with arcs of radius 13 cm from each endpoint. How far is each arc crossing from the midpoint of the segment?
Answer: A
The crossing, the midpoint and an endpoint form a right triangle with hypotenuse 13 and leg 12, so the other leg is √(13² - 12²) = √25 = 5 cm. Choice B is half the segment. Choice C is the radius. Choice D subtracts 13 - 12 = 1 from 12, a misuse of the Pythagorean Theorem.
Question 4 of 20 · Multiple Choice
Why is the line through the two arc crossings the perpendicular bisector of AB?
Answer: B
Each crossing point is one radius from A and the same radius from B. By the converse of the perpendicular bisector theorem, both points lie on the perpendicular bisector, and two points determine the line. Choice A is not a reason. Choice C is false: the arcs are centered at different points. Choice D is false: the line does not pass through A or B.
Question 5 of 20 · Multiple Choice
A student bisects a 124° angle with a compass and straightedge. What does each of the two new angles measure?
Answer: B
An angle bisector divides the angle into two congruent angles: 124° ÷ 2 = 62°. Choice A is the original angle. Choice C bisects twice. Choice D is the supplement, 180° - 124° = 56°.
Question 6 of 20 · Multiple Choice
In the angle bisector construction of Diagram 1, why is ∠XVZ ≅ ∠YVZ?
Answer: C
VX = VY (one arc from V), XZ = YZ (equal arcs from X and Y) and VZ is shared, so the triangles are congruent by SSS and the angles at V match. Choice B gives only one pair of sides. Choice A is irrelevant. Choice D is false: Z lies beyond segment XY, not at its midpoint, and it would not be the reason anyway.
Question 7 of 20 · Multiple Choice
When you copy ∠A to a new vertex A', you first draw arcs of the same radius at A and at A'. Which distance must you copy next with the compass?
Answer: D
Copying the chord between the two crossing points fixes the opening of the angle. Together with the two equal radii, it makes the triangles congruent by SSS. Choice A gives no new information. Choice B relates the two vertices, not the angle. Choice C is not needed: the sides of an angle are rays with no length.
Question 8 of 20 · Multiple Choice
Which congruence criterion explains why the copy-an-angle construction produces an angle congruent to the original?
Answer: A
The construction copies three distances: two equal radii and the chord between the crossing points. Three pairs of congruent sides give SSS. Choices B, C and D each need an angle to be known congruent, which is what the construction is trying to produce.
Question 9 of 20 · Multiple Choice
Point K is 6 cm from line ℓ. An arc of radius 10 cm centered at K crosses ℓ at G and H. What is GH?
Answer: C
The foot of the perpendicular from K is the midpoint of GH. Each half is √(10² - 6²) = √64 = 8 cm, so GH = 16 cm. Choice A is only half of GH. Choice B is the diameter of the circle. Choice D is 10 - 6.
Question 10 of 20 · Multiple Choice
To construct the line through P parallel to ℓ, a student draws a transversal through P meeting ℓ at Q. What should the student do next?
Answer: A
Copying the angle at Q to P as a corresponding angle makes the corresponding angles congruent, so the new line is parallel to ℓ by the converse of the corresponding angles theorem. Choice B produces a line through Q, not P. Choice C gives a line through the midpoint of PQ, not through P, and it is perpendicular to PQ, so it is parallel to ℓ only if PQ ⊥ ℓ. Choice D copies a length, not a direction.
Question 11 of 20 · Multiple Choice
Which paper-folding action constructs the perpendicular bisector of segment AB?
Answer: B
Folding A onto B is a reflection that maps A to B, and the line of reflection is the perpendicular bisector of AB. Choices A and C produce line AB itself. Choice D gives the middle of the paper, which is the bisector only by chance.
Question 12 of 20 · Multiple Choice
A student uses a reflective device (a Mira) to construct the perpendicular bisector of AB. How should the device be placed?
Answer: D
When the image of A falls on B, the device is showing the reflection that maps A to B, and the drawing edge lies on the line of reflection, the perpendicular bisector. Choice A reflects AB onto itself. Choice B gives a line through A. Choice C replaces the construction with measuring.
Question 13 of 20 · Multiple Choice
An angle is drawn on patty paper. Which fold bisects it?
Answer: A
Folding one side onto the other is a reflection that swaps the two sides, so the crease makes congruent angles with them: it is the angle bisector. Choice B moves the vertex, so the crease misses it. Choice C does not create a new ray. Choice D is not possible for most angles and is not a bisector.
Question 14 of 20 · Multiple Choice
One way to construct a parallel line through P is to construct the perpendicular to ℓ through P, and then the perpendicular to that new line at P. Why is the final line parallel to ℓ?
Answer: B
ℓ and the final line both make 90° corresponding angles with the first perpendicular, so they are parallel by the converse of the corresponding angles theorem. Choice A is true of many lines. Choice C is false: perpendicular lines intersect. Choice D is not what the construction shows.
Question 15 of 20 · Short Answer
Describe how to copy segment AB onto a ray from C with a compass and straightedge, and explain why CD = AB. Then explain how to do the same with a string and a pushpin.
Draw a ray from C. Set the compass points on A and B, then without changing the opening, put the point on C and draw an arc crossing the ray at D. CD = AB because D lies on a circle centered at C with radius AB. With string: pinch the string at A and B to fix the length, pin one end at C, and swing the other end to mark D on the ray. The string keeps a fixed length just as the compass keeps a fixed opening.
Question 16 of 20 · Short Answer
List the steps to copy ∠M to a ray from N, and explain why the copied angle is congruent to ∠M.
With any opening, draw an arc centered at M crossing the sides at P and Q. Draw the same arc centered at N, crossing the ray at P'. Set the compass to PQ and draw an arc centered at P' crossing the second arc at Q'. Draw ray NQ'. Then MP = NP' and MQ = NQ' (same radius) and PQ = P'Q' (copied chord), so △MPQ ≅ △NP'Q' by SSS and ∠M ≅ ∠N.
Question 17 of 20 · Short Answer
Segment AB is 6 cm long. Arcs with a 5 cm opening are drawn from A and from B and cross at P and Q. Find PQ, and explain why line PQ crosses AB at its midpoint.
The crossing point, the midpoint and an endpoint form a right triangle with legs 3 and h and hypotenuse 5, so h = √(5² - 3²) = 4 cm, and PQ = 8 cm. P and Q are each 5 cm from A and from B, so both lie on the perpendicular bisector of AB (converse of the perpendicular bisector theorem). That line is PQ, and a perpendicular bisector passes through the midpoint.
Question 18 of 20 · Short Answer
A student bisects a 76° angle and then bisects one of the two halves. What angles are constructed? Explain why the first bisector is exact.
The first bisector makes two 38° angles, and bisecting one of them makes two 19° angles. The first bisector is exact because the arc from the vertex gives two equal distances, the equal arcs from those points give two more equal distances, and the shared ray gives a third pair, so the two triangles are congruent by SSS and the two angles at the vertex are congruent.
Question 19 of 20 · Short Answer
Describe how to construct the line through a point P parallel to a line ℓ by copying an angle, and justify the construction with a theorem.
Draw any line through P that meets ℓ at Q. At Q, draw an arc crossing ℓ and the transversal. Copy that angle at P, on the same side of the transversal and in the corresponding position, using the same radius and chord. Draw the line through P and the new point. The corresponding angles are congruent, so the new line is parallel to ℓ by the converse of the corresponding angles theorem.
Question 20 of 20 · Short Answer
Explain how to construct the perpendicular to a line m through a point K on m, first by paper folding and then with compass and straightedge. Why do the two methods give the same line?
Folding: fold the paper so that m lands on itself and the crease passes through K. Compass: draw an arc centered at K crossing m at G and H, then with a larger opening draw equal arcs from G and H that cross at L, and draw KL. Both give the perpendicular bisector of a segment of m centered at K: the fold reflects G onto H, and L and K are each equidistant from G and H. There is only one line through K perpendicular to m, so the two lines match.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does HSG.CO.D.12 mean?
HSG.CO.D.12 means students must be able to make six basic geometric constructions with several tools. The constructions are copying a segment, copying an angle, bisecting a segment, bisecting an angle, constructing perpendicular lines (including a perpendicular bisector) and constructing a parallel line through a point. "Formal" means the result is exact because of the method, not because of measuring.
What tools does HSG.CO.D.12 allow?
The standard lists compass and straightedge, string, reflective devices, paper folding and dynamic geometry software, and adds "etc.". A ruler scale and a protractor are not construction tools, because reading a scale is measuring. They are useful only to check a finished construction.
What is the difference between drawing and constructing?
A drawing uses measurements, such as marking 4 cm with a ruler. A construction uses only tools that copy distances or reflect figures, so it is exact in theory, and each step has a geometric reason. The arcs left on the page show that the figure was constructed.
Why does the compass opening have to stay the same?
Because most constructions depend on two distances being equal. When you bisect a segment, the arcs from both endpoints must have the same radius, so the crossing points are equidistant from the endpoints. If the opening slips, that equality and the reason behind the construction are lost.
How do you construct a perpendicular bisector by folding?
Fold the paper so that one endpoint lands exactly on the other and crease it. The fold is a reflection that maps one endpoint to the other, and the line of any such reflection is the perpendicular bisector. Patty paper or tracing paper makes this easy to see.
Why do constructions work?
Each one rests on a theorem. Copying a segment uses the definition of a circle. Bisecting a segment and constructing perpendiculars use the converse of the perpendicular bisector theorem. Copying and bisecting an angle use SSS congruence. Constructing a parallel line uses the converse of the corresponding angles theorem.
What is a common mistake in constructions?
Changing the compass opening between arcs that must be equal, or using an opening that is too small to bisect a segment (it must be more than half the segment). Another common mistake is erasing the arcs, which removes the evidence of the construction.
Is HSG.CO.D.12 taught in Geometry?
Yes. It is usually taught in high school Geometry, often alongside the proofs of HSG.CO.C.9, since those theorems explain why the constructions work. HSG.CO.D.13, constructing an equilateral triangle, a square and a regular hexagon inscribed in a circle, builds on it.
Can students use geometry software instead of a compass?
Yes, the standard names dynamic geometry software as one of the tools. Students should use its circle, line and intersection tools the way they would use a compass and straightedge, not its built-in perpendicular or parallel buttons, so that they still carry out the construction. Dragging the points afterwards shows whether the construction holds.
How can parents help with constructions at home?
Ask your student to show a construction with household materials: fold a sheet of paper to find the middle of a line, or use a piece of string and a thumbtack as a compass. Then ask why it works. A good answer names equal distances or a fold that acts as a reflection.
07
Related Standards
5 standards
These standards connect to HSG.CO.D.12: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
HSG.CO.A.1Prerequisite
Know precise definitions of angle, circle, perpendicular and parallel lines, and segment
