Class 10 Mathematics – Triangles | Interactive Visual Revision Notes

📝 Section 1 — Classroom Notes

🔢 Topic 1: Similar Figures

🌎 Real World Connection: Architects draw blueprints where 1 cm on paper = 100 cm in real life — every room is a similar shape scaled down. Surveyors measure heights of mountains by standing far away and using triangle similarity, without ever climbing them. Map-makers use similar triangles to represent vast distances on a small page. Similar triangles are the hidden tool behind every scale drawing.

Two shapes are similar if they look the same.
They have the same shape but can differ in size.
All circles are similar to each other.
All squares are similar to each other.
But all rectangles are NOT similar.

🔢 Topic 2: Similar Triangles

Two triangles are similar if their angles match.
And their sides are in proportion.
We write: △ABC ~ △DEF
The ~ symbol means "is similar to".
Order of letters matters (matching corners).

🔢 Topic 3: Basic Proportionality Theorem (BPT)

Also called Thales' Theorem.
If a line is parallel to one side of a triangle...
...it divides the other two sides in equal ratio.
If DE ∥ BC in △ABC, then:
AD/DB = AE/EC

🔢 Topic 4: Converse of BPT

If a line divides two sides in equal ratio...
...then that line is parallel to the third side.
If AD/DB = AE/EC, then DE ∥ BC.
This is the reverse of BPT.
Very useful for proving lines are parallel.

🔢 Topic 5: AA Similarity Criterion

If two angles of one triangle equal two angles of another...
...the triangles are similar.
We only need two angles to match.
The third angle will automatically match.
This is the easiest way to prove similarity.

🔢 Topic 6: SSS Similarity Criterion

If all three sides are in the same ratio...
...the triangles are similar.
AB/DE = BC/EF = AC/DF
All three ratios must be equal.
No need to check angles.

🔢 Topic 7: SAS Similarity Criterion

If one angle is equal...
...and the sides around it are in same ratio...
...the triangles are similar.
The equal angle must be between the two sides.
This is called the included angle.

🔢 Topic 8: Areas of Similar Triangles

If two triangles are similar:
Ratio of areas = square of ratio of sides.
Area(△1) / Area(△2) = (side1/side2)²
If sides are in ratio 2:3, areas are in ratio 4:9.
Area grows much faster than sides.

🔢 Topic 9: Pythagoras Theorem

In a right triangle:
Hypotenuse² = Base² + Height²
The hypotenuse is the longest side.
It is opposite the 90° angle.
Written as: c² = a² + b²

🔢 Topic 10: Converse of Pythagoras Theorem

If c² = a² + b² in any triangle...
...then the angle opposite c is 90°.
This proves a triangle is right-angled.
Check: 5² = 3² + 4² → 25 = 25 ✔
So 3, 4, 5 form a right triangle.

🎨 Section 2 — Easy Illustrations

Picture 1 — Similar means same shape; Congruent means same size too

Picture 2 — A line parallel to one side divides the other two sides in equal ratio

Picture 3 — AA, SSS, or SAS — any one is enough

Picture 4 — The most famous theorem in mathematics

Picture 5 — When sides double, the area becomes four times

✏️ Section 3 — Worked Examples

✏️ Example 1: Using BPT (Thales' Theorem)

In △PQR, line ST ∥ QR. PS = 4 cm, SQ = 6 cm, PT = 5 cm. Find TR.

1 By BPT: PS/SQ = PT/TR

2 Put the values: 4/6 = 5/TR

3 Cross multiply: 4 × TR = 6 × 5

4 4 × TR = 30

5 TR = 30 ÷ 4 = 7.5 cm

🎯 TR = 7.5 cm

✏️ Example 2: Finding a Side Using Pythagoras

A right triangle has base = 8 cm and hypotenuse = 17 cm. Find the height.

1 c² = a² + b²

2 17² = a² + 8²

3 289 = a² + 64

4 a² = 289 − 64 = 225

5 a = √225 = 15 cm

🎯 Height = 15 cm

✏️ Example 3: Area Ratio of Similar Triangles

△ABC ~ △DEF. If AB = 6 cm, DE = 9 cm, and area of △ABC = 48 cm², find area of △DEF.

1 Side ratio = AB/DE = 6/9 = 2/3

2 Area ratio = (side ratio)² = (2/3)² = 4/9

3 Area(ABC)/Area(DEF) = 4/9

4 48/Area(DEF) = 4/9

5 Area(DEF) = 48 × 9 ÷ 4 = 108 cm²

🎯 Area of △DEF = 108 cm²

📊 Section 4 — Cheat Sheet

Concept	Key Point	Formula / Rule
Similar	Same shape, any size	Angles equal, sides in ratio
Congruent	Same shape + same size	All sides and angles equal
BPT (Thales)	Line ∥ to a side	AD/DB = AE/EC
AA Criterion	2 angles equal	Triangles are similar
SSS Criterion	All 3 side ratios equal	Triangles are similar
SAS Criterion	2 sides in ratio + included angle	Triangles are similar
Area ratio	For similar triangles	(Side ratio)²
Pythagoras	Right triangle only	c² = a² + b²
Converse of Pythagoras	If c² = a² + b²	Triangle is right-angled
Pythagorean Triplets	Whole number sets	(3,4,5), (5,12,13), (8,15,17)

📄 Section 5 — Take-Home Handout

✍️ A. True or False

1. All squares are similar. ___
2. All triangles are similar. ___
3. BPT applies when DE ∥ BC. ___
4. Pythagoras works for all triangles. ___
5. If side ratio is 3:5, area ratio is 9:25. ___

✍️ B. Fill in the Blanks

1. In a right triangle, c² = ________
2. AA stands for ________
3. BPT is also called ________ Theorem
4. Similar triangles have equal ________
5. Area ratio = (________)²

✍️ C. Solve These

1. A right triangle has sides 9 cm and 40 cm. Find the hypotenuse.

Show your work here

2. In △ABC, DE ∥ BC. AD = 3, DB = 5, AE = 4.5. Find EC.

Show your work here

🔗 D. Match the Columns

Column A	Column B
1. BPT	a. c² = a² + b²
2. Pythagoras	b. Same shape, same size
3. Congruent	c. (Side ratio)²
4. Area ratio	d. AD/DB = AE/EC

🎨 E. Check If Right Triangle

Is a triangle with sides 7, 24, 25 a right triangle? Show your work.

Show your work here

✅ Detailed Solutions

A. True or False — Answers

1 TRUE — Similar triangles have equal corresponding angles.

2 FALSE — Similar triangles need NOT be congruent; congruent means same size too.

3 TRUE — BPT states that a line parallel to one side divides the other two in equal ratio.

4 FALSE — AA (Angle-Angle), not AAA, is the similarity criterion (two angles suffice).

5 TRUE — In similar triangles, area ratio = (side ratio)².

B. Fill in the Blanks — Answers

1 In a right triangle, hypotenuse² = a² + b² (Pythagoras theorem).

2 Two triangles are similar by Angle Angle (AA) criterion if two angles match.

3 Thales’ theorem states a line parallel to one side divides the other sides proportionally.

4 Similar triangles have equal corresponding angles.

5 In similar triangles △ABC ~ △DEF, the ratio AB/DE equals the side ratio.

C1. Find hypotenuse for legs 9 and 40

1 By Pythagoras: c² = 9² + 40²

2 c² = 81 + 1600 = 1681

3 c = √1681 = 41

Hypotenuse = 41

C2. BPT: AD = 3, DB = 5, AE = 4.5, find EC

1 By BPT (Thales’ Theorem): AD/DB = AE/EC

2 3/5 = 4.5/EC

3 EC = 4.5 × 5/3 = 22.5/3 = 7.5

EC = 7.5

D. Match the Columns — Answers

1 Similar triangles → d (AD/DB = AE/EC — BPT result)

2 Pythagoras → a (c² = a² + b²)

3 Congruent → b (Same shape, same size)

4 Area ratio → c ((Side ratio)²)

E. Is (7, 24, 25) a right triangle?

1 Check if the largest side squared = sum of other two squares.

2 7² + 24² = 49 + 576 = 625

3 25² = 625

4 Since 7² + 24² = 25², by the converse of Pythagoras, it IS a right triangle.

Yes, (7, 24, 25) is a right triangle ✓

🃏 Section 6 — Revision Flashcards

👉

🔢

What does similar mean?

Same shape but possibly different size. Angles are equal, sides are in proportion.

🔎

What is BPT (Thales' Theorem)?

A line parallel to one side divides the other two sides in equal ratio.

AD/DB = AE/EC

📈

What is AA Criterion?

If two angles of one triangle match two angles of another, they are similar.

❓

State Pythagoras Theorem.

c² = a² + b²

In a right triangle, hypotenuse² = sum of other two sides².

💡

What is the area ratio rule for similar triangles?

Area ratio = (Side ratio)²

If sides are 2:3, areas are 4:9.

🔧

Name three Pythagorean triplets.

(3, 4, 5)

(5, 12, 13)

(8, 15, 17)

🎯

What is the converse of Pythagoras?

If c² = a² + b², then the triangle is right-angled at the vertex opposite c.

🔗

What is SAS Criterion?

If two sides are in ratio and the included angle (angle between those sides) is equal, triangles are similar.

Class 10 Mathematics — Triangles

Original Visual Revision Notes — Different Learner Friendly

Aligned with the Class 10 Mathematics syllabus. Independently created study material. Not affiliated with or endorsed by NCERT.

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

📋 What is Inside?

📝 Section 1 — Classroom Notes

🔢 Topic 1: Similar Figures

🔢 Topic 2: Similar Triangles

🔢 Topic 3: Basic Proportionality Theorem (BPT)

🔢 Topic 4: Converse of BPT

🔢 Topic 5: AA Similarity Criterion

🔢 Topic 6: SSS Similarity Criterion

🔢 Topic 7: SAS Similarity Criterion

🔢 Topic 8: Areas of Similar Triangles

🔢 Topic 9: Pythagoras Theorem

🔢 Topic 10: Converse of Pythagoras Theorem

🎨 Section 2 — Easy Illustrations

✏️ Section 3 — Worked Examples

✏️ Example 1: Using BPT (Thales' Theorem)

✏️ Example 2: Finding a Side Using Pythagoras

✏️ Example 3: Area Ratio of Similar Triangles

📊 Section 4 — Cheat Sheet

📄 Section 5 — Take-Home Handout

✍️ A. True or False

✍️ B. Fill in the Blanks

✍️ C. Solve These

🔗 D. Match the Columns

🎨 E. Check If Right Triangle

✅ Detailed Solutions

A. True or False — Answers

B. Fill in the Blanks — Answers

C1. Find hypotenuse for legs 9 and 40

C2. BPT: AD = 3, DB = 5, AE = 4.5, find EC

D. Match the Columns — Answers

E. Is (7, 24, 25) a right triangle?

🃏 Section 6 — Revision Flashcards

🧠 Section 7 — Mind Map