Master Class 10 Maths Chapter 11: Constructions – Line Division, Similar Triangles & Tangents Explained with NCERT Solutions

✍️ Chapter 11: Constructions – Class 10 Maths

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🔍 Concepts Covered

This chapter focuses on geometrical constructions using a compass and straightedge only, without measuring instruments like protractors. Two major types of constructions are covered:

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🧠 1. Division of a Line Segment in a Given Ratio (m:n)

➤ Concept:

Divide a given line segment internally in the ratio $m:n$ using only ruler and compass.

📌 Construction Steps:

Let’s divide a line segment $AB$ in the ratio $m:n$:

1. Draw $AB$ using a scale.

2. Draw a ray $AX$ making an acute angle with $AB$.

3. Mark $m+n$ equal divisions on $AX$ using compass — say $A_1, A_2, ..., A_{m+n}$.

4. Join $A_{m+n}$ to $B$.

5. Draw a line through $A_m$ parallel to $A_{m+n}B$ (using construction of a triangle with equal corresponding angles).

6. This line meets $AB$ at point $P$.
   ✅ Hence, $P$ divides $AB$ in the ratio $m:n$.

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🧠 2. Construction of Similar Triangles Using Scale Factor (k = m:n)

➤ Concept:

To construct a triangle similar to a given triangle using a scale factor $m:n$.

🔁 Cases:

• If $m > n$, constructed triangle is smaller than the original.

• If $m < n$, constructed triangle is larger than the original.

📌 Construction Steps (e.g., scale factor $4:3$):

1. Construct the given triangle $ABC$.

2. Extend base $BC$.

3. Draw a ray from point $B$ making an acute angle with $BC$.

4. Mark 4 points (m) on this ray using compass.

5. Join 3rd point (n) to $C$.

6. Draw a line through the 4th point parallel to the line joining 3rd point and $C$.

7. Repeat similar steps for remaining triangle sides using parallel lines.

8. The resulting triangle is the required one.

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🧠 3. Construction of Tangents to a Circle from an External Point

➤ Concept:

From a point $P$ outside a circle with center $O$, draw two tangents to the circle.

📌 Construction Steps:

1. Draw circle with center $O$ and radius $r$.

2. Mark point $P$ outside the circle such that $OP > r$.

3. Join $OP$.

4. Find midpoint $M$ of line segment $OP$.

5. Draw a circle with center $M$ and radius $MO$.

6. Let it intersect the original circle at $Q$ and $R$.

7. Join $PQ$ and $PR$.
   ✅ These are the required tangents.

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📘 NCERT Exercise Solutions

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✅ Exercise 11.1 – Solutions

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Q1. Divide a line segment of length 10 cm internally in the ratio 3:2.

Solution:

• Draw a line $AB = 10$ cm.

• Draw ray $AX$ making an acute angle.

• Mark 5 equal divisions on $AX$ using compass (3+2 = 5).

• Label them as $A_1, A_2, ..., A_5$.

• Join $A_5$ to $B$.

• Draw a line through $A_3$ parallel to $A_5B$ meeting $AB$ at $P$.

✅ Hence, point $P$ divides $AB$ in ratio 3:2.

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Q2. Construct a triangle similar to a given triangle as per the given scale factor.

Example: Construct a triangle similar to triangle $ABC$, scale factor $\frac{4}{3}$ (triangle larger than original).

Steps:

1. Draw triangle $ABC$.

2. Extend base $BC$.

3. Draw ray $BX$ making acute angle with $BC$.

4. Mark 4 points on $BX$, join 3rd to $C$.

5. Draw parallel line from 4th point to intersect extended side.

6. Join and complete triangle $A'B'C'$ similar to $ABC$.

✅ $\triangle A'B'C' \sim \triangle ABC$

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✅ Exercise 11.2 – Solutions

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Q1. Draw a circle of radius 4 cm. From a point 6 cm away from the center, draw two tangents to the circle.

Solution:

• Draw a circle with center $O$ and radius 4 cm.

• Take point $P$ such that $OP = 6$ cm.

• Join $OP$.

• Find midpoint $M$ of $OP$.

• Draw circle with center $M$, radius $MO$.

• Let it intersect the original circle at $Q$ and $R$.

• Join $PQ$ and $PR$.

✅ These are the required tangents from $P$ to the circle.

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📋 CBSE Previous Year Questions from Constructions (with year)

Year	Question	Marks
2020	Draw tangents to a circle of radius 4 cm inclined at an angle of 60°	3
2019	Draw tangents from a point 6.5 cm away from center of circle (r = 3.5 cm)	3
2018	Divide a 7 cm line in the ratio 3:4	3
2016	Construct a triangle similar to a given triangle with ratio 3:5	3
2015	From a point 7 cm from the center of a circle of radius 3 cm, draw tangents	4

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📌 Key Theorems/Terms

• Tangent to a circle: A line that touches the circle at exactly one point and is perpendicular to the radius at the point of contact.

• Similar Triangles: Two triangles are similar if their corresponding angles are equal and their corresponding sides are in the same ratio.

• Scale Factor: Ratio $\frac{m}{n}$ used to construct similar triangles.

• Midpoint Theorem (used indirectly): In tangent constructions, midpoint helps in forming right triangle via semicircle.

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📚 Summary

Topic	Techniques Used
Line segment division	Basic geometry, parallel lines
Similar triangles	Proportionality, basic construction
Tangents	Right triangles, midpoint circle, compass usage

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