Key Rules and Methods for Triangle Construction
Concept of Triangles
We all are already aware of the concept of Triangle. A Triangle may be a three-sided Polygon made from Three sides having three angles. It is to be noted that within the construction of a triangle the three sides and angle may or might not be equal in dimensions.
Practical geometry, which deals with the development of various geometrical figures, is a crucial branch of Geometry. Using a set of geometrical tools such as rulers, compasses and protractors, different shapes like squares, triangles, circles, hexagons, etc. can be constructed. The only condition is that you should be aware of the properties of these figures that set them apart from one another. You are already aware of the construction of lines, angles, bisectors, etc. This knowledge is extended for triangle construction. But before we move to that, in this section, we bring to you all the properties of triangles that you need to keep in mind before moving over to triangle construction.
Example 1:-
Construct a triangle when the bottom , one base angle and the sum of the lengths of the opposite two sides are given
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Draw the bottom BC and at the point B , B makes an angle, say XBC, adequate to the given angle.
Cut a line segment BD adequate to AB + AC from the ray BX.
Join DC and make an angle DCY adequate to BDC.
Let CY intersect BX at A
ABC is the required triangle.
In triangle ACD, ∠ACD = ∠ ADC
So, we can write AB = BD – AD = BD – AC
AB + AC = BD
Example 2:-
Now for constructing a triangle whose perimeter and both base angles are given, the first step is to:
Draw base of any length
Draw base of any length = Perimeter
To draw the base angles from the given or random line
Draw the base with length = 1/3 × perimeter
Solution: B is the correct option. For constructing a triangle whose perimeter and base angles both have been given, the first step is to draw the base of any length = Perimeter.
Draw Different Type of Triangles -
Construction of a Congruent Triangle :-
We're working on a neighborhood with four different models, and this is our first one. We want to copy it, which means making a congruent triangle. Thank goodness we live in a two-dimensional geometry world, or we'd probably need to know things about plumbing and electrical work.
Let's start with a point. Let's call it D. Now, draw a ray using the ruler from D to form our base, or foundation of the house (see video starting at 01:00 to see these actions). Next, take the compass and measure the distance from A to C. Use this width to draw an arc that hits our ray. Where it hits is point F, the equivalent of point C on the model.
Next, use the compass to measure the distance from A to B. Again, use this to draw an arc around where the top of the new triangle should be. Don't add a point yet. We're not sure exactly where the top of our house will be.
Let's measure C to B and draw another arc, this time from point F. Where these arcs meet is our final point, point E. Now, just connect D to E and F to E, and we have a congruent triangle.
SSS Construction of Triangles
Construct ΔXYZ during which XY = 4.5 cm, YZ = 5 cm and ZX = 6cm.
Step 1. Draw a line YZ of length 5 cm.
Step 2. From point Y, the point X is at a distance of 4.5 cm. So, with Y as the center of the line, draw an arc of radius 4.5 cm.
Step 3. From point Z, point X is at a distance equal to 6 cm. So, with Z as the center of the line, draw an arc of radius 6 cm.
Step 4. X has got to get on both the arcs drawn. So, it is the point of intersection of arcs. Mark the point of interaction of arcs as X. Join XY and XZ. ΔXYZ is the required triangle.
Construction of an isosceles triangle
A triangle may be a triangle with two equal side lengths and two equal angles. Sometimes you'll have to draw a triangle given limited information. If you know the side lengths, base, and altitude, it is possible to do this with just a ruler and compass (or just a compass, if you are given line segments). Using a protractor, you'll use information about angles to draw a triangle .
Case where AB > AC or AC > AB
We have been given the base of a triangle, the base angle of the triangle and difference of the other two sides of the triangle.
Now for constructing the triangle (∆ABC) such that base BC,the base angle∠B and difference of the other two sides (that is AB – AC or AC-AB )is given to us, then for constructing such triangles these two cases can arise:
Length of AB > Length of AC
Length of AC > Length of AB
The Following Steps of Construction are Followed for the Given Two Cases -
Here are the Steps of Construction if AB > AC:
1. First draw the base BC of the triangle ∆ABC as given and construct XBC of the required measure at B as shown below .
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2. Now from the ray BX ,cut an arc that is equal to AB – AC at point P and join it to C as shown
3. Now , draw the perpendicular bisector of PC and let it intersect at the point BX at point A .
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4.When we join AC, we get ∆ABC which is the required triangle.
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Focus on the Steps of Construction if AC > AB:
1. Draw the base BC of ∆ABC as given and construct XBC of the required measure at point B as shown below:
(Image to be added soon)
2. From the given ray BX you need to cut an arc equal to AB – AC at point P and join it to C. In this case the point P will lie on the opposite side to the ray BX. Now draw the perpendicular bisector of PC and now let it intersect BX at point A as shown below-
(Image to be added soon)
3. Now when we will join the points A and C, the triangle we get as ∆ABC is the required triangle.
FAQs on Construction of Triangles Made Easy
1. What are the essential geometric tools required for constructing triangles?
To accurately construct triangles as per the CBSE curriculum for the 2025-26 session, you need a few essential tools. These are:
A ruler (or straightedge) to draw line segments of precise lengths.
A compass to draw arcs and circles of specific radii, which is crucial for marking lengths and locating vertices.
A protractor to measure and draw angles accurately.
A sharp pencil and an eraser for clear and neat constructions.
2. What are the four main criteria that guarantee the construction of a unique triangle?
A unique triangle can be constructed if we have specific sets of measurements. The four main criteria, also known as congruence rules, are:
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 angle between them are known.
ASA (Angle-Side-Angle): The measures of two angles and the length of 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 using a ruler.
Step 2: Set the compass to the length of the second side. Place the compass point on vertex A and draw an arc.
Step 3: Set the compass to the length of the third side. Place the compass point on vertex B and draw another arc that intersects the first one.
Step 4: Label the intersection point as C. Join C to A and C to B to complete the required triangle ABC.
4. What is the method 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:
Step 1: Draw a base line segment, for example, PQ, with the length of one of the given sides.
Step 2: At one of the endpoints (say, P), use a protractor to draw a ray that makes an angle equal to the given included angle.
Step 3: Use a compass to measure the length of the second given side. With P as the centre, cut an arc on the ray drawn in the previous step. Label this point R.
Step 4: Join R to Q using a ruler. The resulting triangle PQR is the required triangle.
5. What is the process for constructing a triangle if two angles and the included side are known (ASA criterion)?
The construction of a triangle using the ASA criterion involves these steps:
Step 1: Draw a line segment, say YZ, equal to the length of the given included side.
Step 2: At endpoint Y, use a protractor to draw a ray making one of the given angles.
Step 3: At endpoint Z, use the protractor to draw another ray making the second given angle.
Step 4: The point where these two rays intersect is the third vertex, X. Triangle XYZ is the required triangle.
6. Why can't any three random line segments form a triangle? What is the rule behind this?
Not any three line segments can form a triangle due to a fundamental rule called the Triangle Inequality Theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. For example, you cannot construct a triangle with sides 3 cm, 4 cm, and 8 cm because the sum of the two smaller sides (3 + 4 = 7) is less than the third side (8). The two shorter sides would not be long enough to meet and form a vertex.
7. How is the ASA (Angle-Side-Angle) criterion different from the AAS (Angle-Angle-Side) criterion in construction?
The key difference lies in the position of the known side. In the ASA criterion, the given side is the one included between the two known angles. In the AAS criterion, the given side is a non-included side. For construction purposes, this difference is manageable. If you are given two angles of a triangle, you can always find the third by subtracting their sum from 180°. This means an AAS case can be converted into an ASA case before starting the construction, allowing you to use the standard ASA method.
8. How is constructing a right-angled triangle using the RHS criterion different from using other criteria like SAS or SSS?
The RHS (Right angle-Hypotenuse-Side) criterion is a special case exclusively for right-angled triangles. It requires knowing the right angle (90°), the side opposite to it (the hypotenuse), and one other side. This is different from:
SAS: This would require knowing the two sides that form the 90° angle (the legs), not the hypotenuse.
SSS: This would require knowing all three sides, including both legs and the hypotenuse.
The RHS criterion is unique because it allows construction without knowing both legs or all three sides, leveraging the fixed 90° angle and the unique properties of the hypotenuse.
9. In what real-world applications is the skill of constructing triangles important?
The principles of triangle construction are fundamental in many fields beyond the classroom. Key examples include:
Architecture and Engineering: Triangles are the most stable geometric shape. Engineers use them to design strong structures like bridges (trusses), roofs, and towers.
Navigation and Surveying: A technique called triangulation uses triangles to determine the precise location of points by measuring angles from known distances.
Computer Graphics and Animation: Complex 3D models in video games and movies are built from a mesh of thousands of tiny triangles (polygons).
Art and Design: Artists use principles of composition based on triangles to create balance and direct the viewer's eye.
10. How can you construct a special triangle, like one with angles 30°, 60°, and 90°?
Constructing a 30°-60°-90° triangle can be done without a protractor by using a compass and ruler:
Step 1: Draw a line segment AB. Construct a 60° angle at point A by drawing an arc and then, from the intersection point on AB, cutting that arc with the same compass width.
Step 2: Bisect the 60° angle. This involves drawing two new arcs from the points where the first arc intersected the arms of the 60° angle. The line joining A to this new intersection point will form a 30° angle.
Step 3: Extend the lines for the 30° angle and the original base. To form the 90° angle, you can drop a perpendicular from any point on one arm to the other. The resulting triangle will have angles measuring 30°, 60°, and 90°.
