Essential Steps to Draw a Triangle Using a Compass
The way of constructing a triangle with a compass depends on the information given in the question. The construction of triangles is very important while applying the Pythagorean theorem and trigonometry.
Here, we will learn how to construct a triangle if we have the following data.
All the three sides of a triangle are given.
The measure of the hypotenuse and one side is given in the right triangle.
Two sides of a triangle and included angle are given.
Two angles of a triangle and included sides are given.
How to Construct Triangles?
Triangle is a two-dimensional polygon shape with three sides and three angles, which can be formed by joining the points in a plane.
But, the question arises how to construct triangles?
A unique triangle can be easily constructed using the concept of Geometry.
Geometry is a branch of Mathematics that deals with lines, angles, shapes, size, and dimension of different things we observe in everyday life. In Euclidean Geometry, there are different two dimensional and three -dimensional shapes. Flat shapes such as square, triangle, and circle are known as two -dimensional shapes. These shapes have only length and width.
Solid shapes such as cube, cuboid, sphere, cone, etc are three-dimensional shapes. These shapes have length, width, or height.
These geometric shapes can be easily constructed using compass, ruler, and protractor. Let us learn the steps of constructing triangles with compass, ruler, and protector below.
Constructing Triangle When Hypotenuse and One Side is Given
To construct a triangle when hypotenuse and one side is given, we need the following geometric tools:
A Ruler
A Compass
Let us learn to construct a triangle when hypotenuse and one side is given through examples:
Construct a right-angled triangle ABC with the length of the hypotenuse AB = 3 cm and side BC = 5 cm. The steps of construction are:
Step 1:
Draw a line of any length and mark a point C on it.
Step 2:
Set the width of the compass to 3 cm.
Step 3:
Place the pointer of the compass at C and draw an arc on both sides of C.
Step 4:
Mark the point as P and A where both the arc crosses the line.
Step 5:
Taking P as the centre, draw an arc above the point C.
Step 6:
Taking A as the centre, draw an arc that cuts the previous arc.
Step 7:
Mark the point B, where two arcs intersect each other.
Step 8:
Join the points B and A along with B and C with the help of the ruler.
Thus, ΔABC is the required right-angled triangle.
Constructing Triangle When Two Sides and Included Angle are Given
To construct a triangle when two sides and angle, we need the following geometric tools:
A Ruler
A Protractor
A Compass
Let us learn to construct a triangle when the length of two sides and included angle are given through an example.
Example:
Construct a triangle PQR with PQ = 4 cm, QR = 6.5 cm , and ∠PQR = 60°.
The steps of construction are:
Step 1:
Draw a line QR = 6.5 cm using a ruler.
Step 2:
Using protractor at Q, draw a line QX making an angle of 60° with QR
Step 3:
Taking Q as the centre, draw an arc of radius 4 cm to cut the line QX at P.
Step 4:
Join PR.
Therefore, PQR is the required triangle.
Constructing Triangle When Two Sides and Included Angle are Given
To construct a triangle when two sides and angle, we need the following geometric tools:
A Ruler
A Protractor
Let us learn to construct a triangle when the length of one side and included angle are given through examples.
Construct a triangle XYZ with XY = 4 cm, ∠ZXY = 100° and ∠ZXY = 30°.
The Steps of Construction are:
Step 1:
Draw a line segment XY = 4 cm using a ruler
Step 2:
Using protractor at X, draw a ray XP forming an angle of 30° with XY
Step 3:
Using protractor at Y, draw another ray YQ making an angle of 100° with XY
Step 4:
Let the rays XP and QY intersect at Z.
Step 5:
Using the property, sum of all the angles of a triangle is equal to 180°, we can easily find the third angle of the triangle which is 50°. Hence, ∠Z = 50°.
Step 6:
Hence, XYZ is a required triangle.
Constructing Triangle Given Three Sides
To construct a triangle when all the three sides are given, we need the following geometric tools:
A Ruler
A Protractor
Before knowing how to construct a triangle with given sides, we should check the following property of triangles is met by the length of all the three sides.
“ The sum of all the three sides of a triangle should always be greater than its third side”.
We will not be able to construct a triangle with the given three sides if the above-mentioned property is not met by the given three sides.
Let us learn to construct a triangle given three sides through an example.
Example:
Construct a triangle ABC with side AB = 4 cm, BC = 6 cm and AC = 5 cm.
The steps of construction are:
Step 1:
Draw a line BC = 6 cm ( the longest side).
Step 2:
Taking B as centre, draw an arc of radius 4 cm above the line segment BC.
Step 3:
Taking C as centre, draw an arc of radius 5 cm that intersects the previous arc at ‘A’.
Step 4:
Join line segments AB and AC
Hence, ABC is the required triangle.
Drawing Triangle With Protractor
Construct an isosceles triangle PQR with PQ = 6 cm, QR = 6 cm and ∠PQR = 50°.
Steps of drawing a triangle with protractor for the given sides and angles are as follows:
Draw a line QR 6 cm long.
Taking Q as the centre, draw an angle of 50° using the protractor.
Taking R as the centre, draw an angle of 50° using the protractor (angles opposite to the equal sides of an isosceles triangle are equal).
Mark the point P where two lines intersect.
Therefore, PQR is the required isosceles triangle.
Solved Examples:
1. Construct an equilateral triangle with a side 5 cm long using a protractor?
Ans: An equilateral triangle is a triangle whose all the three sides are equal in length. Another property of the equilateral triangle is that three angles of the triangle are equal, and each angle of a triangle is equal to 60 degrees.
Following are steps to construct an equilateral triangle with each side 5 cm long.
Step 1:
Draw a line AB of 5 cm long.
Step 2:
Taking A as centre, draw an angle of 60° using a protractor.
Step 3:
Taking B as centre, draw another angle of 60° using a protractor.
Step 4:
Mark the point C where both the lines meet.
Hence, ABC is a required equilateral triangle of length 5 cm.
2. Write down the steps in constructing a triangle ABC with sides AB = 3.5 units, BC = 6 units and AC = 4.5 units.
Solution
Step 1:
Draw a line segment BC measuring 6 units.
Step 2:
With B as center, and draw an arc of radius 3.5 units
Step 3:
With C as center, draw an arc of radius 4.5 units to intersect the previous arc at A
Step 4:
Join the line segment AB and AC.
Hence, the triangle ABC is drawn.
Fun Facts
Triangle is a polygon with the minimum possible number of sides (three).
Hatch marks, also known as tick marks are used in triangles to identify the sides of equal length.
Two triangles are considered similar if each angle of one triangle has the same measure as the corresponding angle in the other triangle.
FAQs on How to Construct a Triangle With a Compass
1. What are the essential tools from a geometry box required to construct a triangle?
To accurately construct a triangle, you primarily need three essential tools from your geometry box. These are:
A ruler (or straightedge): Used to draw straight line segments of specific lengths.
A compass: Used to draw arcs and circles of a set radius, which is crucial for marking lengths and locating vertices.
A protractor: Used to measure and draw angles to a specific degree.
2. What are the four main congruence criteria that allow for the construction of a unique triangle?
A unique triangle can be constructed if you have one of the following four sets of measurements, which correspond to the triangle congruence criteria:
SSS (Side-Side-Side): The lengths of all three sides are known.
SAS (Side-Angle-Side): The lengths of two sides and the measure of the included angle (the angle between them) are known.
ASA (Angle-Side-Angle): The measures of two angles and the length of the included side (the side between them) are known.
RHS (Right-angle-Hypotenuse-Side): In a right-angled triangle, the length of the hypotenuse and one other side are known.
3. How do you construct a triangle when the lengths of all three sides are given (SSS criterion)?
To construct a triangle using the SSS criterion, follow these steps:
Step 1: Draw a line segment, say AB, equal to the length of one of the given sides.
Step 2: Open the compass to the length of the second side. With point A as the centre, draw an arc.
Step 3: Now, open the compass to the length of the third side. With point B as the centre, draw another arc that intersects the first arc at a point. Label this point C.
Step 4: Join points A to C and B to C. The resulting triangle, ABC, is the required triangle.
4. What is the step-by-step process for constructing a triangle with two given sides and the included angle (SAS criterion)?
To construct a triangle using the SAS criterion, you need to follow this process:
Step 1: Draw a line segment which will be the base of the triangle, for instance, AB, with one of the given lengths.
Step 2: Place the protractor at point A and draw the given included angle.
Step 3: Use your compass or ruler to mark the length of the second given side along the new line from point A. Let's call this point C.
Step 4: Join point C to point B to complete the triangle ABC.
5. Why is it impossible to construct a triangle if the sum of two sides is less than or equal to the third side?
This is due to a fundamental property called the Triangle Inequality Theorem. For any three points to form a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. If you try to construct it, the arcs you draw from the endpoints of the base side will never intersect. Visually, the two shorter sides are not long enough to meet and form a third vertex, so they will either fall short or lie flat on the longest side.
6. What is the important difference between constructing a triangle using SAS versus the SSA condition?
The key difference is uniqueness. The SAS (Side-Angle-Side) criterion guarantees the construction of one unique triangle because the given angle is fixed between the two known sides. However, the SSA (Side-Side-Angle) condition, where the angle is not included between the sides, does not guarantee a unique triangle. Depending on the given lengths, SSA can result in two possible triangles, one triangle, or no triangle at all. This ambiguity is why SSA is not a standard criterion for constructing a unique triangle.
7. How is the principle of triangle construction applied in real-world scenarios?
The principles of triangle construction are fundamental in many fields for ensuring stability and measuring distances. Key examples include:
Architecture and Engineering: Triangles are the strongest simple geometric shape. They are used in the construction of bridge trusses, roof supports, and geodesic domes to distribute weight and stress effectively.
Surveying and Navigation: A technique called triangulation is used to determine the location of a point by measuring angles to it from known points at either end of a fixed baseline, essentially constructing a triangle to find distances.
Computer Graphics: 3D models in video games and animations are created from a mesh of thousands of tiny triangles (polygons) that define their shape and surface.
8. How can you construct a special right-angled triangle with angles 90°, 60°, and 30° using only a compass and ruler?
You can construct a 30-60-90 triangle by leveraging basic angle constructions:
Step 1: Start by constructing an equilateral triangle, as all its angles are 60°.
Step 2: Choose one vertex and bisect its 60° angle. To do this, place the compass on the vertex, draw an arc that crosses the two sides, and then draw intersecting arcs from those two points. The line from the vertex through this intersection splits the 60° angle into two 30° angles.
Step 3: The line you drew for the angle bisector will also be the perpendicular bisector of the opposite side, meeting it at a 90° angle. This divides the equilateral triangle into two identical 30°-60°-90° right-angled triangles.
