NCERT Solutions Class 10 Maths Chapter 11 teaches you precise geometric constructions using only compass and ruler. You’ll learn how to divide line segments in given ratios (like 3:4), construct tangents to circles from external points, and draw triangles similar to given triangles with specific scale factors. Each solution shows the exact construction steps with proper arc markings and justifications, helping you score full marks in the 4-6 mark construction questions that appear in every CBSE board exam.
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All exercises with step-by-step solutions | Updated 2025-26 | Free Download
Download PDF (Free)NCERT Solutions Class 10 Maths Chapter 11 Constructions – Complete Guide
NCERT Class 10 Chapter 11 – Constructions builds upon your foundational geometry skills from Class 9, introducing you to more advanced construction techniques that carry 5 marks in your CBSE board examination. This chapter focuses on two main types of constructions: dividing a line segment in a given ratio (internally and externally) and constructing tangents to a circle from an external point. These constructions form the backbone of practical geometry and have real-world applications in architecture, engineering design, and technical drawing.
📊 CBSE Class 10 Maths Chapter 11 – Exam Weightage & Marking Scheme
| CBSE Board Marks | 5 Marks |
| Unit Name | Geometry |
| Difficulty Level | Medium |
| Importance | Medium |
| Exam Types | CBSE Board, State Boards |
| Typical Questions | 1-2 questions |
You’ll explore step-by-step procedures for dividing line segments using the basic proportionality theorem, which connects directly to your understanding of similar triangles from Chapter 6. The construction of tangents to circles reinforces your knowledge of circle properties from Chapter 10, making this chapter an excellent integration of multiple geometric concepts. Each construction comes with a mathematical justification, helping you understand not just the ‘how’ but also the ‘why’ behind each step.
For your CBSE board exam, expect 1-2 questions from this chapter, typically worth 3-5 marks. Questions usually ask you to perform a specific construction and justify the steps, or to construct and then measure certain elements. The difficulty level is medium, but with regular practice, you can score full marks as the steps are standardized and follow logical patterns. Many students find this chapter scoring because constructions either follow the correct procedure or they don’t—there’s little ambiguity in marking.
Quick Facts – Class 10 Chapter 11
| 📖 Chapter Number | Chapter 11 |
| 📚 Chapter Name | Constructions |
| ✏️ Total Exercises | 1 Exercises |
| ❓ Total Questions | 14 Questions |
| 📅 Updated For | CBSE Session 2025-26 |
Mastering constructions requires hands-on practice with actual compass and ruler. While understanding the theory is important, your success depends on practicing each construction multiple times until you can execute them accurately and neatly. Focus on maintaining proper arc marks, clear labeling, and writing concise justifications. With dedication and practice, you’ll develop the precision and confidence needed to excel in geometric constructions and secure those valuable marks in your board examination.
NCERT Solutions Class 10 Maths Chapter 11 – All Exercises PDF Download
Download exercise-wise NCERT Solutions PDFs for offline study
| Exercise No. | Topics Covered | Download PDF |
|---|---|---|
| Exercise 11.1 | Complete step-by-step solutions for 14 questions | 📥 Download PDF |
Constructions – Key Formulas & Concepts
Quick reference for CBSE exams
| Formula | Description | When to Use |
|---|---|---|
| Dividing a Line Segment Internally \[AM/MB = m/n\] | If point M divides line segment AB internally in the ratio m:n, then AM/MB equals m/n. Note: M lies *between* A and B. The ratio is between the lengths of the two resulting segments. | To find the location of a point that divides a line segment in a given ratio. |
| Construction of Similar Triangles (Scale Factor > 1) \[A’B’ = k * AB\] where k > 1 | If triangle A’B’C’ is similar to triangle ABC with a scale factor k greater than 1, then each side of A’B’C’ is k times the corresponding side of ABC. Note: Extend the sides of the original triangle to get the larger similar triangle. | To construct a triangle larger than a given triangle, similar to it with a given scale factor. |
| Construction of Similar Triangles (Scale Factor < 1) \[A’B’ = k * AB\] where k < 1 | If triangle A’B’C’ is similar to triangle ABC with a scale factor k less than 1, then each side of A’B’C’ is k times the corresponding side of ABC. Note: The new triangle will be inside the original triangle. | To construct a triangle smaller than a given triangle, similar to it with a given scale factor. |
| Drawing a Tangent to a Circle from a Point on it \[OT \perp PT\] | The tangent at any point of a circle is perpendicular to the radius through the point of contact. Therefore, angle OTP is 90 degrees, where O is the center, T is the point on the circle, and P is a point on the tangent. Note: Use the property that the radius and tangent are perpendicular. Construct a perpendicular to the radius at the given point. | To construct a tangent line to a circle at a specific point on the circle. |
| Drawing a Tangent to a Circle from a Point Outside it \[PA = PB\] | The lengths of tangents drawn from an external point to a circle are equal. PA and PB are tangents from point P to a circle with center O. Note: Find the midpoint of the line joining the external point and the center. Draw a circle with this midpoint as center and radius equal to half the length of the line joining the external point to the centre. The points where this circle intersects the given circle will be the points from where tangents can be drawn. | To construct tangents to a circle from a point outside the circle. You’ll use the bisection of the line joining the center of the circle and the external point. |
| Steps for Dividing a Line Segment N/A | Dividing a line segment in a given ratio m:n – Step 1: Draw a ray AX making an acute angle with AB. Step 2: Mark m+n points A1, A2, … Am+n on AX such that AA1 = A1A2 = … = Am+n-1Am+n. Step 3: Join BAm+n. Step 4: Through the point Am, draw a line parallel to Am+nB intersecting AB at P. Then AP:PB = m:n. Note: Remember to draw the parallel line accurately using a compass and ruler. | To divide a line segment internally in a given ratio. |
| Angle Sum Property \(\angle A + \angle B + \angle C = 180^{\circ}\) | The sum of the angles of a triangle is 180 degrees. This is important when trying to create a triangle with specific angles. Note: Make sure the angles you construct sum up to 180 degrees. | When constructing a triangle with given angles. Verifying if a constructed triangle satisfies given angle requirements. |
| Basic Proportionality Theorem (Thales’ Theorem) \[\frac{AD}{DB} = \frac{AE}{EC}\] | If a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides the two sides in the same ratio. Note: The line DE must be parallel to BC in triangle ABC for this theorem to be applicable. D lies on AB, E lies on AC. | To understand the relationship between sides when constructing similar triangles. Useful in verification of construction. |
Frequently Asked Questions – NCERT Class 10 Maths Chapter 11
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