👇 📐 Class 10 Maths Chapter 11: Constructions – Notes

 

🛠️ Introduction

In earlier classes, you learned how to construct triangles and angles using a compass and ruler. In Class 10, you’ll expand that knowledge to more advanced constructions, including:

• Dividing a line segment in a given ratio

• Constructing tangents to a circle

• Constructing similar triangles

All constructions must be done precisely using geometric tools (no protractors).

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📏 1. Division of a Line Segment in a Given Ratio

🧮 Objective:

To divide a given line segment $AB$ in the ratio $m:n$ using only a compass and straightedge.

✍️ Steps:

1. Draw line segment $AB$.

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

3. Mark $m + n$ equal divisions on $AX$ using compass (say, A1, A2,...).

4. Join the last point (say $A_{m+n}$) to point B.

5. Draw a line from $A_m$ parallel to $A_{m+n}B$, meeting AB at point P.

6. Point P divides AB in the ratio $m:n$.

✅ Construction Complete!

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🔺 2. Construction of Similar Triangles

Objective:

To construct a triangle similar to a given triangle with a given scale factor (greater than or less than 1).

Case 1: Scale factor k > 1 (enlargement)

1. Construct triangle $ABC$.

2. Extend base side $AB$.

3. On the extended ray, divide into k equal parts.

4. Mark the k-th point and draw a line parallel to BC from the last division point.

5. Complete triangle.

Case 2: Scale factor k < 1 (reduction)

Follow similar steps, but mark divisions before the original point.

📌 Tip: Use parallel lines and AA similarity to guide the construction.

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🟢 3. Construction of Tangents to a Circle

Case 1: From a point on the circle

1. Draw the radius to the point.

2. Draw a perpendicular to the radius at that point – that’s the tangent.

Case 2: From a point outside the circle

1. Join the external point $P$ to the center $O$.

2. Find the midpoint of $OP$.

3. Draw a circle with the midpoint as center and radius = $\frac{OP}{2}$.

4. This circle intersects the original circle at two points.

5. Join those points to $P$ – these are the two tangents.

✅ You’ve constructed two tangents from an external point!

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📝 Sample Constructions

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

Use the division method described above.

Q2. Construct a triangle similar to a given triangle with scale factor 3:2.

Use the enlargement or reduction method.

Q3. Construct tangents to a circle of radius 4 cm from a point 7 cm away.

Apply the external point tangent construction steps.

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📌 Key Points to Remember

• Use only compass and ruler (no angle measurers).

• Practice constructions neatly and label all points clearly.

• For triangle similarity: parallel lines and ratios matter.

• For tangents: always perpendicular to radius at point of contact.

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📥 Download Chapter 11 PDF Notes: Coming Soon

📘 Previous Chapter: Chapter 10 – Circles »
📚 Next Chapter: Chapter 12 – Areas Related to Circles »