How to Perform Geometric Constructions Step by Step with Solved Examples
An Overview of CBSE Class 9 Maths Constructions
Geometric constructions class 9 walks you through the steps of how different geometrical shapes like triangle, polygons, circle, etc. are drawn with the help of a compass and ruler. The scope of constructions in Maths for class 9 introduces students to the bisection of angles, construction of a perpendicular, etc. and is considered vital for solving problems based on geometry.
You can refer to CBSE class 9 maths constructions solutions to understand the steps of construction and its approach more effectively. In turn, it will also help you to solve problems based on Constructions Class 9 NCERT PDF easily.
Read on to find more about the constructions class 9 CBSE chapter!
Constructions Class 9 NCERT – A Brief Overview
For geometrical constructions, mostly two instruments are used – a non-graduated ruler and a compass. Before moving to the concepts covered in the chapter, you need to become familiar with the geometrical tools. You can check class 9 maths constructions NCERT solutions for more details about geometry instruments.
A. Components of Geometry Box
Typically, a geometry box comprises these important instruments –
A Graduated Scale
This instrument comes in handy for drawing straight lines. One of the sides of the graduated scale is marked in ‘cm’ and ‘mm’, whereas; the other side is marked in inches.
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Set Squares
This instrument comprises one set-square with angles 30°, 60° and 90°. On the other hand, the second set comprises angles 45° and 90°. Solve problems on constructions class 9 Maths and find out the use of set squares in practice.
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Compass
A compass comes in handy for constructing circles and different angles. It comes with a provision to fit a pencil at the instrument’s end. Refer to solved examples to find out more about how to do construction class 9 with a compass.
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Divider
A divider proves useful for measuring lengths accurately. Check out constructions class 9 ex 11.2 to find out if you need to use a divider or not.
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Protractor
It is useful in measuring and marking angles accurately. In fact, NCERT solutions of constructions class 9 elaborate the requirement of a protractor for solving constructions class 9 exercise 11.2 and other exercises.
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With the help of these tools, you will be able to solve construction class 9 extra questions smoothly and will finish exercises like the construction of class 9 exercise 11.1 in no time.
B. Basic Construction for Class 9 CBSE
Some of the basic constructions covered in class 9 maths chapter 11 constructions include –
i. Construction of an Angle Bisector
Step 1 – Take B as the centre and proceed to draw an arc of a specific radius intersecting BC and BA. Name the intersecting points as E and D.
Step 2 – Taking D and E as its centre draw arcs that intersect each other at a point ‘F’ which will make a radius more than ½ of DE.
Step 3 – Then, a line BF has to be drawn, which will serve as the required bisector of the angle ABC.
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Solve the construction of class 9 exercise 11.2 to find similar problems and solved examples.
ii. Construction of a 60° Angle
You may come across problems in construction chapter class 9 exercise 11.1, which will require you to construct a 60° angle.
Step 1 – Draw a line QR.
Step 2 – Taking Q as the centre, construct an arc with any radius. Mark Y as the intersecting point of QR.
Step 3 – Without changing the radius, take point Y as the centre and draw an arc to intersect the previous arc at a point X.
Step 4 – Draw a line QP passing through point X. This will make the required angle PQR.
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Find a detailed explanation of similar problems in NCERT solutions for class 9 maths chapter 11 study rankers and gain a better understanding of the approach.
iii. Construction of Triangles Class 9 ICSE and CBSE
(Highlighting the base angle and summation of the two sides of a triangle)
Suppose, in a triangle ABC,
BC is the base
Angle B is the base
Summation of the sides (AB+AC) is given.
Step 1 – Draw the base of a triangle BC.
Step 2 – Construct angle B to make XBC.
Step 3 – A line segment BD has to be drawn, which will make BD = AB+AC on the line BX.
Step 4 – DC has to be joined to make the angle DCY = BDC.
Step 5 – Intersect BX with CY at point A, making ABC the required triangle.
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Check out constructions class 9 solutions to learn how to construct triangles and solve problems on them with ease. You will also find answers to class 9 maths constructions extra questions in the study solutions along with an adequate explanation.
iv. Construction of a Triangle when Perimeter and 2 Base Angles are Known
Suppose the required triangle is ABC
Step 1 – The line segment XY = AB+BC+CA is drawn.
Step 2 – Make angle MYX = angle C.
Step 3 – Make angle LYX = angle B.
Step 4 – Angle LYX and MYX is intersected at a point A.
Step 5 – PQ will intersect XY at B and RS will intersect XY at C.
Step 6 – Join AC and AB, making ABC the required Triangle.
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Refer to constructions class 9 NCERT solutions to find step by step explanation about each geometric pattern. Learn more about the construction of polygons class 9 ICSE and other important shapes under the guidance of subject experts. Join our live online classes and get all your doubts cleared and pick up effective tips to solve NCERT maths exercise 11.1 of class 9 effectively.
To gain a better idea about construction chapter class 9, download our class 9 maths exercise 11.2 solutions and develop an effective approach towards the chapter and its exercises.
Download our Vedantu App now to access constructions class 9 notes!
FAQs on Constructions in Geometry with Compass and Ruler
1. What are constructions in geometry?
Geometric constructions are accurate drawings made using only a compass and straightedge without measuring lengths or angles. These constructions are used to create precise figures based on given conditions.
- No ruler markings are used to measure distances.
- Only arcs and straight lines are drawn.
- Common constructions include bisecting angles, drawing perpendicular lines, and constructing triangles.
2. What tools are used in geometric constructions?
The main tools used in geometric constructions are a compass and a straightedge (ruler without markings). These tools allow precise geometric drawing.
- Compass: Draws arcs and circles.
- Straightedge: Draws straight lines between points.
3. How do you bisect a line segment?
To bisect a line segment, you draw arcs from both endpoints with the same radius and join their intersection points to find the midpoint. Follow these steps:
- Given line segment AB, place the compass at A and draw an arc above and below the line.
- With the same radius, repeat from point B.
- Join the two arc intersection points with a straight line.
4. How do you bisect an angle in constructions?
An angle is bisected by drawing equal arcs from the angle’s arms and joining their intersection to the vertex to form the angle bisector. Steps:
- Draw an arc centered at the vertex cutting both arms of the angle.
- From those two intersection points, draw arcs that intersect each other.
- Join the vertex to the arc intersection point.
5. How do you construct a perpendicular bisector?
A perpendicular bisector is constructed by drawing equal arcs from each endpoint of a segment so that they intersect above and below the line. Steps:
- Given segment AB, draw arcs from A with radius more than half of AB.
- With the same radius, draw arcs from B.
- Join the arc intersection points.
6. How do you construct a triangle with given sides?
A triangle with given three sides (SSS construction) is made by drawing one side and using arcs to locate the third vertex. Steps:
- Draw base AB equal to one given side.
- With center A, draw an arc equal to the second side.
- With center B, draw an arc equal to the third side.
- The intersection of arcs gives point C.
7. What is the difference between drawing and construction in geometry?
The key difference is that construction uses only a compass and straightedge without measuring, while drawing may use rulers, protractors, and measurements.
- Construction is based on geometric properties.
- Drawing may involve estimation or measurement tools.
- Constructions are logically justified step by step.
8. How do you construct a perpendicular line from a point on a line?
To construct a perpendicular from a point on a line, draw equal arcs on the line and use them to form a 90° line through the point. Steps:
- Let P be the point on line l.
- Draw an arc centered at P cutting the line at A and B.
- With centers A and B and equal radius, draw arcs intersecting above the line.
- Join P to the arc intersection point.
9. How do you construct a circle passing through three points?
A circle passing through three non-collinear points is constructed by finding the intersection of the perpendicular bisectors of two sides. Steps:
- Join the three points to form a triangle.
- Construct perpendicular bisectors of any two sides.
- Their intersection is the circumcenter.
- Draw a circle centered at this point passing through one vertex.
10. Why are geometric constructions important in maths?
Geometric constructions are important because they develop understanding of geometric properties, logical reasoning, and accuracy.
- They explain why shapes have certain properties.
- They are used in triangle construction and circle theorems.
- They build foundations for coordinate geometry and proofs.
