Class 10 Mathematics – Constructions | Interactive Visual Revision Notes

📝 Section 1 — Classroom Notes

🌎 Real-World Connection: When an architect divides a plot of land in a specific ratio for two owners, or when a civil engineer draws a road parallel to another at a fixed distance, they apply geometric construction principles. A tailor dividing cloth in exact proportions, or a designer scaling a drawing — these real tasks rely on the same compass-and-ruler methods you learn in this chapter.

📐 Topic 1: What Is a Construction?

A construction uses ruler and compass only.
We do not measure angles with protractor.
We draw exact shapes, not rough sketches.
Constructions give precise geometric figures.
All steps must be shown with arcs.

📐 Topic 2: Basic Tools Needed

A ruler draws straight lines.
A compass draws arcs and circles.
A sharp pencil keeps lines thin.
A protractor is used only to verify.
Keep compass tight so it does not slip.

📐 Topic 3: Dividing a Line Segment

We can divide a line in any ratio.
Example: divide AB in ratio 3 : 2.
We use the Basic Proportionality Theorem.
Also called Thales theorem method.
No measurement needed, only compass arcs.

📐 Topic 4: Steps to Divide a Line

Draw line AB of the given length.
Draw a ray at any angle from A.
Mark equal arcs on the ray (m + n arcs).
Join last arc point to B.
Draw a parallel line through the m-th point.

📐 Topic 5: Constructing Similar Triangles

Similar triangles have same shape, different size.
All angles stay the same.
All sides change by the same ratio.
This ratio is called scale factor.
We write scale factor as a fraction.

📐 Topic 6: Scale Factor Greater Than 1

Scale factor > 1 means bigger triangle.
Example: scale factor 5/3 means enlarge.
New sides = old sides × 5/3.
Mark 5 arcs on the ray (larger number).
The new triangle is outside the old one.

📐 Topic 7: Scale Factor Less Than 1

Scale factor < 1 means smaller triangle.
Example: scale factor 2/3 means shrink.
New sides = old sides × 2/3.
Mark 3 arcs on the ray (larger number).
The new triangle is inside the old one.

📐 Topic 8: Tangent to a Circle

A tangent touches the circle at one point.
It is perpendicular to the radius there.
The angle between radius and tangent = 90 degrees.
We use this fact to construct tangents.
Draw radius, then a line at right angle.

📐 Topic 9: Two Tangents from External Point

From a point outside, draw two tangents.
Both tangents have equal length.
Find the midpoint of line joining center and point.
Draw a circle with this midpoint as center.
Where circles meet = tangent points.

📐 Topic 10: Tips for Accurate Constructions

Use a sharp pencil always.
Do not change compass width mid-step.
Draw arcs long enough to intersect clearly.
Label all points as you go.
Keep construction lines light and thin.

🎨 Section 2 — Easy Illustrations

Picture 1: Dividing a Line in Ratio 3 : 2

Draw a ray, mark 5 equal arcs, join last to B, draw parallel from 3rd arc to find P.

Picture 2: Similar Triangle (Scale Factor 5/3)

Scale factor 5/3 means each side becomes 5/3 times the original. The triangle grows bigger.

Picture 3: Similar Triangle (Scale Factor 2/3)

Scale factor 2/3 means each side becomes 2/3 of the original. The triangle shrinks.

Picture 4: Tangent from a Point on the Circle

The tangent at point T is always at 90 degrees to the radius OT.

Picture 5: Two Tangents from an External Point

Find midpoint M of OP. Draw circle with center M passing through O and P. Where it meets the original circle gives T1 and T2.

✏️ Section 3 — Worked Examples

Example 1: Divide a Line Segment in a Given Ratio

Question: Divide a line of 7.5 cm in ratio 2 : 3.

1 Draw line AB = 7.5 cm.

2 Draw a ray AX at any angle from A.

3 Total parts = 2 + 3 = 5 parts.

4 Mark 5 equal arcs on ray AX: A1, A2, A3, A4, A5.

5 Join A5 to B.

6 From A2, draw line parallel to A5B.

7 This line meets AB at point P.

8 AP = 7.5 × 2/5 = 3 cm. PB = 7.5 × 3/5 = 4.5 cm.

P divides AB so that AP : PB = 2 : 3

Check: 3 + 4.5 = 7.5 cm ✓

Example 2: Construct a Similar Triangle

Question: Triangle has sides 4, 6, 8 cm. Construct similar with scale factor 3/2.

1 Draw the original triangle with sides 4, 6, 8 cm.

2 Scale factor = 3/2. Since 3 > 2, triangle is bigger.

3 Draw a ray from one vertex at any angle.

4 Mark 3 arcs on the ray (the larger of 3, 2).

5 Join the 2nd arc to the corresponding vertex.

6 From the 3rd arc, draw a parallel line.

7 New sides: 4 × 3/2 = 6, 6 × 3/2 = 9, 8 × 3/2 = 12 cm.

New triangle has sides 6, 9, and 12 cm

All sides multiplied by 3/2 ✓

Example 3: Two Tangents from an External Point

Question: Circle has radius 4 cm. Point is 10 cm from center. Draw two tangents. Find tangent length.

1 Draw circle with center O, radius = 4 cm.

2 Mark point P at 10 cm from O.

3 Find midpoint M of OP. OM = MP = 5 cm.

4 Draw circle with center M, radius 5 cm.

5 This circle meets the original circle at T1 and T2.

6 Join PT1 and PT2. These are the tangents.

7 Tangent length: PT = √(OP² − OT²)

8 PT = √(100 − 16) = √84 = 2√21 ≈ 9.17 cm

Each tangent length = √84 = 2√21 ≈ 9.17 cm

Formula: tangent = √(distance² − radius²) ✓

📊 Section 4 — Cheat Sheet

Term / Formula	What It Means	Key Fact
Construction	Drawing with ruler and compass	No protractor used
Divide Line	Split line in ratio m : n	Mark m + n arcs
Scale Factor	Ratio of new to old sides	> 1 bigger, < 1 smaller
Similar Triangle	Same shape, different size	All angles equal
Tangent	Line touching circle at one point	Perpendicular to radius
Tangent Length	√(d² − r²)	d = distance, r = radius
Two Tangents	From external point	Both are equal in length
Right Angle	Radius meets tangent	Always 90 degrees
Midpoint Circle	Center = midpoint of OP	Finds tangent points
BPT	Basic Proportionality Theorem	Used to divide lines

📄 Section 5 — Take-Home Handout

📝 A. Fill in the Blanks

1. A construction uses only _________ and _________.
2. To divide a line in ratio 4 : 3, mark _________ arcs.
3. A tangent is _________ to the radius at the point of contact.
4. Scale factor greater than 1 makes a _________ triangle.
5. Two tangents from an external point are _________ in length.

✓ B. True or False

1. A tangent can touch a circle at two points. _______
2. Scale factor 3/4 gives a smaller triangle. _______
3. We need a protractor for constructions. _______
4. The angle between radius and tangent is 90 degrees. _______
5. Two tangents from an outside point have different lengths. _______

✏️ C. Solve These

1. Divide a line of 9 cm in the ratio 3 : 4. Write all steps.

Show your work here

2. A circle has radius 5 cm. A point is 13 cm from the center. Find the tangent length.

Show your work here

🔗 D. Match the Columns

Column A	Column B
1. Scale factor 5/2	a. Perpendicular to radius
2. Tangent line	b. Bigger triangle
3. Divide line in 3 : 5	c. √(d² − r²)
4. Tangent length	d. Mark 8 arcs

🎨 E. Practice Construction

Draw a circle of radius 3 cm. Mark a point 8 cm from the center. Construct two tangents from that point to the circle.

Draw your construction here

✅ Detailed Solutions

Solution to Q1: Dividing a Line Segment in Ratio 3:4

Question: Divide a line segment AB of length 9 cm in the ratio 3:4 using construction.

1 Draw line segment AB = 9 cm.

2 Draw a ray AX making an acute angle with AB. Mark 3 + 4 = 7 equal arcs on AX, labelling them A₁, A₂, …, A₇.

3 Join A₇ to B. Draw a line through A₃ parallel to A₇B, meeting AB at point P.

4 P divides AB in the ratio 3:4. Verify: AP = (3/7) × 9 = 27/7 ≈ 3.86 cm.

5 PB = (4/7) × 9 = 36/7 ≈ 5.14 cm. Check: 3.86 + 5.14 = 9 cm ✓

∴ AP = 27/7 ≈ 3.86 cm | PB = 36/7 ≈ 5.14 cm

Solution to Q2: Tangent from External Point to a Circle

Question: A point P is 13 cm from the centre of a circle of radius 5 cm. Find the length of the tangent from P to the circle.

1 Let the tangent touch the circle at T. Then OT = 5 cm (radius) and OP = 13 cm.

2 OT ⊥ PT (radius ⊥ tangent). Use Pythagoras: PT² = OP² − OT².

3 PT² = 13² − 5² = 169 − 25 = 144.

4 PT = √144 = 12 cm.

∴ Length of tangent = 12 cm

🃏 Section 6 — Revision Flashcards

👉

📐

What tools do we use in a construction?

Only a ruler and a compass. No protractor or set square.

✂

How do you divide a line in ratio m : n?

Draw a ray from one end. Mark m + n equal arcs. Join last arc to other end. Draw parallel from the m-th arc.

🔼

What is a scale factor?

The ratio of the sides of the new triangle to the original. Example: 3/2 means new side = old side × 3/2.

🔺

Scale factor greater than 1 means?

The new triangle is bigger than the original. Example: 5/3 makes each side 5/3 times longer.

🔻

Scale factor less than 1 means?

The new triangle is smaller than the original. Example: 2/3 makes each side 2/3 times shorter.

⭕

What angle does a tangent make with the radius?

Always 90 degrees. The tangent is perpendicular to the radius at the point of contact.

📏

How do you find tangent length from an external point?

Tangent = √(d² − r²)

d = distance from center, r = radius.

📌

Are the two tangents from an external point equal?

Yes! Both tangents from an external point to a circle are always equal in length.

Class 10 Mathematics — Constructions

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📐 Class 10 Mathematics — Constructions Interactive Visual Revision Notes Aligned with the Class 10 Mathematics syllabus. Independently created study material. Not affiliated with or endorsed by NCERT.

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