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Constructions

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visibility 113update 8 months agobookmarkshare

🎯 In this topic you will

  • Construct triangles accurately using given dimensions
  • Draw the perpendicular bisector of a line segment
  • Draw the bisector of an angle
  • Construct angles of 60°, 45°, and 30°
  • Use a circle to draw a regular polygon
 

🧠 Key Words

  • arc
  • bisector
  • construct (geometry)
  • hypotenuse
  • inscribe
Show Definitions
  • arc: A part of the circumference of a circle between two points.
  • bisector: A line or ray that divides an angle or segment into two equal parts.
  • construct (geometry): To accurately draw shapes, lines, or angles using only a compass, ruler, or straightedge.
  • hypotenuse: The longest side of a right-angled triangle, opposite the right angle.
  • inscribe: To draw a shape inside another so that all vertices touch the boundary of the outer shape.
 

🟠 Constructing Triangles

You need to be able to draw a triangle when you know some of the sides and angles.

You can do this using computer software. You can also do it using a ruler and compasses.

Here are four different examples of how to construct triangles.

 

🔺 Constructing a Triangle Using ASA

When you know two angles and the side between them, this is known as ASA.

Step 1: Draw the side.
Draw an angle at one end.

Step 2: Draw the angle at the other end.
Where the two lines cross is the third vertex of the triangle.

Step 2 ASA triangle
 

🟠 Constructing a Triangle Using SAS

When you know two sides and the angle between them, this is known as SAS.

Step 1: Draw the angle first.

Step 1 SAS triangle
Step 2: Open your compasses to $12\ \text{cm}$. Put the point of the compasses on A and draw an arc to mark B. Mark C in a similar way. Draw the side BC.

Step 2 SAS triangle
Tip
An arc is part of a circle.
 

🟠 Constructing a Triangle Using SSS

When you know the three sides but no angles, this is known as SSS.

Triangle with SSS known

Step 1: Draw one side.
Open your compasses to the length of a second side.
Put the point of the compasses on one end of the side and draw an arc.

Step 1 SSS triangle
Step 2: Open your compasses to the length of a second side.
Put the point of the compasses on the other end of the side and draw another arc.
Where the arcs cross is the third vertex. Draw the other two sides.

Step 2 SSS triangle
 

🟠 Constructing a Triangle Using RHS

When one angle is a right angle, and you know the length of the hypotenuse and one other side, this is known as RHS.

Step 1: Draw the side.
Draw a right angle at one end.

Step 1 RHS triangle
Step 2: At the other end, draw an arc equal to the hypotenuse.
Draw the third side.

Step 2 RHS triangle
Tip
The hypotenuse is the side opposite the right angle.
 

🟠 Constructing the Bisector of a Line Segment

There are two other constructions that you need to be able to do using a ruler and compasses:

1 Construct the bisector of a line segment.

This is a line through the mid-point of the line segment and perpendicular to it.

Step 1: Draw the line segment. Open the compasses to about the same length as the line. (You do not need to measure this exactly.)
Draw arcs from one end of the line on both sides of the line.

Step 1 bisecting line segment
Step 2: Do the same thing at the other end of the line segment.
Do not change the angle between the arms of the compasses.

Step 2 bisecting line segment
 

🟠 Constructing the Bisector of an Angle

Construct the bisector of an angle.
This is a line that divides an angle into two equal parts.

Step 1: Open the compasses to a few centimetres.
You do not need to measure this.
Put the point of the compasses on the angle and draw arcs that cross each of the lines.

Step 1 bisecting angle
Step 2: Put the compass point on each of the crosses and draw an arc between the two lines.
Do not change the angle between the arms of the compasses.
Draw a line through the angle and the last cross.
This is the perpendicular bisector of the angle. The two angles marked are equal.

Step 2 bisecting angle
Tip
When you do any construction, do not rub out your construction lines.
Draw them faintly and leave them on your drawing.
 

🧠 PROBLEM-SOLVING Strategy

Constructing and Measuring Triangles

Use these steps to accurately draw, label, and measure triangle properties using given angles and sides.

  1. Use a ruler to draw a base side (e.g., $AB = 6\ \text{cm}$).
  2. Use a protractor to measure an angle from a base point (e.g., $\angle CAB = 65^\circ$).
  3. Draw the angled side with a ruler, using the correct length (if given) or extending to intersect the other side.
  4. Label all points clearly: $A, B, C$ or other letters from the problem.
  5. If needed, use a protractor to measure unknown angles.
  6. If only sides are given (e.g., $7\ \text{cm},\ 8.5\ \text{cm},\ 9.7\ \text{cm}$), use a compass to draw arcs and find triangle intersections.
  7. To construct bisectors:
    • For angle bisectors, use a compass to draw equal arcs from the angle vertex.
    • For perpendicular bisectors, draw arcs from each endpoint using the same radius, then connect their intersections.
  8. Use a ruler to measure any unknown side (e.g., $BC$ or $XZ$) and label the measurement.
  9. Write down all results (angles or side lengths) clearly near your diagram.
  10. Double-check all measurements and correct inaccuracies if necessary.
 

❓ EXERCISES

1. Draw an accurate copy of this triangle.

a. Draw an accurate copy of this triangle.

b. Measure the length of $AC$ and $BC$.

👀 Show answer
Use a protractor to draw angles $65^\circ$ and $42^\circ$ from each end of the $6\,\text{cm}$ base. Measure sides $AC$ and $BC$ with a ruler once constructed.

2. 

a. Draw an accurate copy of this triangle.

b. Measure the length of $XY$ and $XZ$.

👀 Show answer
Start with the $5\,\text{cm}$ side $YZ$ and use a protractor to draw $36^\circ$ and $100^\circ$ angles. Measure $XY$ and $XZ$ with a ruler after construction.

3. 

a. Draw an accurate copy of this triangle.

b. Measure angle $Q$.

👀 Show answer
Draw side $PR = 10\,\text{cm}$, then angle $50^\circ$ at $P$. From $P$, use a compass to draw an arc of radius $7\,\text{cm}$ to locate point $Q$. Then measure angle $Q$.

4. 

a. Draw an accurate copy of this triangle.

b. Measure angle $F$.

👀 Show answer
Draw base $DE = 10\,\text{cm}$, then use a protractor to mark angle $117^\circ$ at point $E$. From $E$, draw a $6\,\text{cm}$ line to point $F$. Then measure angle $F$ using a protractor.
 

❓ EXERCISES

5. The hypotenuse of a right-angled triangle is $12.5\,\text{cm}$. One of the other sides is $10\,\text{cm}$.

a. Make an accurate drawing of the triangle.

b. Measure the third side.

c. Measure the other two angles.

👀 Show answer
Start by drawing the $10\,\text{cm}$ side from one end of a right angle. Use compasses to draw a $12.5\,\text{cm}$ arc for the hypotenuse. After drawing the triangle, measure the unknown side and angles with a ruler and protractor.

6. The sides of a triangle are $7\,\text{cm}$, $8.5\,\text{cm}$, and $9.7\,\text{cm}$.

a. Make an accurate drawing of the triangle.

b. Measure the largest angle of the triangle.

👀 Show answer
Use a ruler and compasses to construct a triangle with the given side lengths. After completing the triangle, measure the largest angle using a protractor—it will be opposite the longest side, $9.7\,\text{cm}$.

7. The sides of a triangle are $5.8\,\text{cm}$, $7.8\,\text{cm}$, and $7.1\,\text{cm}$.

a. Make an accurate drawing of the triangle.

b. Give your triangle to a partner to check the accuracy of your drawing.
If necessary, correct your drawing.

👀 Show answer
Use a compass-based method to draw the triangle with sides $5.8\,\text{cm}$, $7.8\,\text{cm}$, and $7.1\,\text{cm}$. Have a classmate verify the side lengths using a ruler, and revise your drawing if errors are found.
 

🧠 Think like a Mathematician

Scenario: Two sides of a right-angled triangle are $10.5\ \text{cm}$ and $8.3\ \text{cm}$. Zara and Arun give different answers for the third side:

  • Zara says the third side is 6.4 cm.
  • Arun says the third side is 13.4 cm.

Task: Use accurate drawings to show that both of them could be correct.

Equipment: Ruler, protractor, compasses, calculator

Method:

  1. Draw a right-angled triangle with sides $10.5\ \text{cm}$ and $8.3\ \text{cm}$ adjacent to the right angle.
  2. Measure the length of the third side. Check if it is approximately 13.4 cm.
  3. Draw another right-angled triangle where $10.5\ \text{cm}$ is the hypotenuse and $8.3\ \text{cm}$ is one of the other sides.
  4. Measure the missing side in this case. Check if it is approximately 6.4 cm.

Follow-up Questions:

1. Why are both Zara and Arun correct even though their answers are different?
2. What does this tell you about the position of the right angle in a triangle?
3. Explain how the Pythagorean Theorem helps justify both constructions.
👀 show answer
  • 1: Both are correct because they placed the right angle in different positions—once between the two sides and once opposite one of them.
  • 2: The position of the right angle determines which side is the hypotenuse. This changes the third side's length.
  • 3: Using the Pythagorean Theorem:
    • For Arun: $\sqrt{10.5^2 8.3^2} \approx 13.4$
    • For Zara: $\sqrt{10.5^2 - 8.3^2} \approx 6.4$
 

❓ EXERCISES

9.

a. Draw this diagram accurately.

b. Construct the perpendicular bisector of $AB$.

c. The perpendicular bisector of $AB$ intersects $AC$ at $D$. Label $D$ on your diagram and measure $AD$.

👀 Show answer
Start by drawing base $AB = 10\,\text{cm}$ and angle $39^\circ$ at point $A$. From $A$, draw side $AC = 7.5\,\text{cm}$. Construct the perpendicular bisector of $AB$ using compass arcs from both $A$ and $B$. Label the intersection with $AC$ as point $D$, and then use a ruler to measure $AD$.
 

🧠 Think like a Mathematician

Scenario:$RST$ is a triangle. $RS = 5 \text{ cm}$, $RT = 6 \text{ cm}$.

Part a: If $ST = 9 \text{ cm}$, use a diagram to show that angle $R$ is obtuse.
Write the size of angle $R$.

Part b: If angle $R$ is obtuse, what can you say about the length of $ST$?
Give reasons for your answer.

👀 show answer
  • a: Use the cosine rule to calculate angle $R$:
    $\cos R = \dfrac{RS^2 RT^2 - ST^2}{2 \cdot RS \cdot RT} = \dfrac{5^2 6^2 - 9^2}{2 \cdot 5 \cdot 6} = \dfrac{61 - 81}{60} = \dfrac{-20}{60} = -\dfrac{1}{3}$
    $\Rightarrow R \approx 109.5^\circ$, which is obtuse.
  • b: Since angle $R$ is obtuse, the side opposite it, $ST$, must be the longest side in the triangle. This confirms that $ST = 9 \text{ cm}$ is indeed longer than $RS$ and $RT$, which makes sense geometrically.
 

❓ EXERCISES

11. Draw this triangle accurately.

a. Construct the bisector of angle $A$.

b. The bisector of angle $A$ intersects $BC$ at $X$.

c. Mark $X$ on your triangle and measure $BX$.

👀 Show answer
a. Use a protractor to draw angle $39^\circ$ between $AB = 10\text{ cm}$ and $AC = 7.5\text{ cm}$.
b. Use a compass to construct the angle bisector of $\angle CAB$.
c. Mark $X$ where the bisector crosses $BC$. Measure $BX$ using a ruler.

12. Read what Marcus and Sofia say.

a. Draw a triangle. Construct the perpendicular bisector of each side to test Marcus’s theory.

b. Look at the triangles that other learners have drawn. Do you think Marcus is correct?

c. Draw another triangle. Construct the bisector of each angle to test Sofia’s theory.

d. Look at the triangles that other learners have drawn. Do you think Sofia is correct?

👀 Show answer
a. Use a compass to construct the perpendicular bisector of each side. They should meet at a single point — this supports Marcus’s claim.
b. Yes, Marcus is correct — the perpendicular bisectors of a triangle intersect at the circumcenter.
c. Construct the angle bisectors of the triangle using a compass. They also meet at a single point — this supports Sofia’s claim.
d. Yes, Sofia is correct — the angle bisectors of a triangle meet at the incenter.
 

🔧 Introduction to Further Constructions

You know how to construct a perpendicular or bisect an angle using a ruler and compasses. In this section, you will learn to do more constructions with a ruler and compasses. Here are three examples.

 
📘 Worked example

Inscribe a square in a circle.

Answer:

a. Draw a circle and a diameter.

b. Construct the perpendicular bisector of the diameter.

c. The points where the diameters meet the circle are the vertices of a square.

Tip. Put the point of the compasses on the end of the diameter.

 
📘 Worked example

Construct an angle of $45^\circ$.

Answer:

Draw a line segment and its perpendicular bisector.

Bisect one of the right angles. This gives two angles of $45^\circ$.

 

 
📘 Worked example

Construct an angle of $60^\circ$.

Answer:

Draw a line segment AB.

Open the compasses to the length of AB.

Draw arcs from A and B.

The point where the arcs cross is C.

Join A to C. Angle $CAB$ is $60^\circ$.

Join B to C. Angle $CBA$ is $60^\circ$.

 

 

🧠 PROBLEM-SOLVING Strategy

Compass-and-Straightedge Constructions

Quick methods to construct key angles and inscribe regular polygons in a circle.

  1. Start with a clean base. Draw a reference line segment AB and a circle (centre O) with convenient radius $r$.
  2. Perpendiculars & bisectors. Use equal-radius arcs from the ends of a segment to get its perpendicular bisector; this gives right angles $90^\circ$ and midpoints for later steps.
  3. Equilateral triangle ⇒ $60^\circ$. With radius $r$, draw arcs from A and B to meet at C. Triangle ABC is equilateral, so angle $\angle CAB = 60^\circ$.
  4. Angle bisection. To halve any angle, draw arcs from the vertex cutting both rays; intersect those arcs and join to the vertex to get the bisector. Examples: $60^\circ \to 30^\circ$, $90^\circ \to 45^\circ$, and $30^\circ \to 15^\circ$.
  5. Regular hexagon from a circle. Step the compass (set to radius $r$) around the circumference from a starting point to mark $6$ equal chords; join adjacent marks for a regular hexagon (each interior angle $120^\circ$).
  6. Square & octagon from diameters. Draw two perpendicular diameters to get a square; add the bisectors of those diameters to obtain $8$ equally spaced points and join in order for a regular octagon (interior angle $135^\circ$).
  7. Dodecagon idea. Halve the central step of a hexagon: mark the six hexagon points, then bisect each central arc to get $12$ points. Join sequentially for a regular dodecagon (interior angle $150^\circ$).
  8. Accuracy checks. In a regular $n$-gon inscribed in a circle of radius $r$: equal chords $=$ equal sides, central step angle $\dfrac{360^\circ}{n}$, and interior angle $\dfrac{(n-2)\times 180^\circ}{n}$. Verify with ruler/protractor only after construction.
Polygon Vertices ($n$) Interior angle
Hexagon $6$ $120^\circ$
Octagon $8$ $135^\circ$
Dodecagon $12$ $150^\circ$

Tip: Keep the compass opening fixed when stepping around a circle; changing it breaks regularity.

 

❓ EXERCISES

1. Construct an angle of $60^\circ$.

a. Construct an angle of $60^\circ$.

b. Bisect your angle in part a to make an angle of $30^\circ$.

c. Use a protractor to check the accuracy of your angles.

👀 Show answer
Use compass and straightedge to create $60^\circ$ as in the worked example, then bisect to get $30^\circ$. Verify with protractor.

2. Draw an equilateral triangle with each side $6\ \text{cm}$ long.

b. Use a ruler and protractor to check the accuracy of your drawing.

👀 Show answer
Draw a triangle with all sides $6\ \text{cm}$. Check that all sides are equal and each angle is $60^\circ$.

3. Draw a circle with a radius of $4\ \text{cm}$.

b. Inscribe a square in the circle.

c. Measure the length of each side of your square.

👀 Show answer
Use the diameter and perpendicular bisector method to draw the square. Measure to confirm equal sides.

4. Construct two perpendicular diameters in a circle.

b. Construct a diameter bisecting each of the diameters in part a. Your diagram should look like this:

c. Join the ends of the diameters to form a regular octagon.

d. What is the interior angle of a regular octagon?

e. Ask a partner to check that your octagon is regular by measuring the sides and angles.

👀 Show answer
Interior angle of a regular octagon is $135^\circ$. Verify equal side lengths and angles to confirm regularity.

5. Construct a triangle with angles $30^\circ$, $60^\circ$ and $90^\circ$.

b. The longest side of your triangle should be double the length of the shortest side. Use this fact to check the accuracy of your drawing.

👀 Show answer
Use the standard geometric construction for a $30^\circ$-$60^\circ$-$90^\circ$ triangle. Measure to confirm longest side is twice the shortest.
 

❓ EXERCISES

6. Construct an equilateral triangle.

b. Use your equilateral triangle to construct a triangle with angles $30^\circ$, $30^\circ$ and $120^\circ$.

👀 Show answer
Construct an equilateral triangle, then adjust side placement to form a triangle with two $30^\circ$ angles and one $120^\circ$ angle.

7. O is the centre of a circle. OAB and OCB are equilateral triangles.

a. Construct a copy of the diagram.

b. Extend the diagram to inscribe a regular hexagon in the circle.

c. What size are the angles of a regular hexagon?

d. Ask a partner to check the accuracy of your construction.

👀 Show answer
A regular hexagon has internal angles of $120^\circ$. Verify equal side lengths and angles to confirm regularity.

8. Use a ruler and compasses to construct angles of

a. $120^\circ$

b. $15^\circ$

👀 Show answer
Use the known constructions: $120^\circ$ from an equilateral triangle and $15^\circ$ by bisecting $30^\circ$.

9. Draw a circle with a radius of $6\ \text{cm}$.

b. Mark point P on the circumference. Put your compass point on P. Draw two arcs on the circumference of radius $6\ \text{cm}$.

c. Draw more arcs on the circumference from these two points. Do not change the angle between the arms of your compasses when you do this.

d. Keep your compasses the same and draw one more arc so you have six points on the circumference.

e. Join the six points to make a hexagon.

f. Check that your hexagon is regular and that the length of each side is $6\ \text{cm}$.

👀 Show answer
A regular hexagon inside the circle has all sides equal to the radius, $6\ \text{cm}$, and internal angles of $120^\circ$.

9. This diagram shows four identical equilateral triangles.

a. Construct a copy of the diagram.

b. This diagram shows four identical triangles with angles $30^\circ$, $60^\circ$ and $90^\circ$. Construct a copy of the diagram.

👀 Show answer
Copy the diagrams exactly using compass and straightedge, ensuring all sides and angles match the given triangle properties.

10. Draw a large circle. Inscribe a regular dodecagon inside the circle.

b. What is the size of each angle of a regular dodecagon? Use this fact to check the accuracy of your drawing.

💡 Tip

A dodecagon has 12 sides.

👀 Show answer
Each internal angle of a regular dodecagon is $150^\circ$. Verify equal side lengths and angles to confirm regularity.
 

❓ EXERCISES

11. This pattern has rotational symmetry of order $6$.

a. i. Construct a copy of the pattern.

a. ii. How did you do the construction? Is there a different way? Which way is better?

This pattern has rotational symmetry of order $4$.

b. i. Construct a copy of the pattern.

b. ii. How did you do the construction? Is there a different way? Which way is better?

👀 Show answer
Use compass and straightedge to replicate each figure by marking equal arc lengths around the circle. Alternative methods may involve symmetry folding or rotational tracing; the better method is the one that maintains accuracy with fewer construction steps.

12. Here are two rhombuses.

a. Construct a copy of each rhombus.

b. Ask a partner to check the accuracy of your drawings.

👀 Show answer
Use the given angle measures ($60^\circ$, $120^\circ$, $30^\circ$, $150^\circ$) and equal side lengths to construct each rhombus accurately. Verify with a ruler and protractor.
 

📘 What we've learned

  • How to use a ruler and compasses to construct standard angles such as $60^\circ$, $90^\circ$, $45^\circ$, $30^\circ$, and $15^\circ$ by combining and bisecting known angles.
  • Methods for inscribing regular polygons (square, hexagon, octagon, dodecagon) inside a circle using perpendicular diameters and equal chord steps.
  • That in a regular $n$-gon, each interior angle is given by $\frac{(n-2)\times 180^\circ}{n}$, and all sides are equal in length.
  • How to verify constructions using a protractor and ruler, checking both side lengths and angle measures for accuracy.
  • The importance of keeping the compass radius fixed when stepping around a circle to ensure regularity of the polygon.

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