1
These are always similar —
💡 Hint:
The shape is the same.
Explanation:
All circles are similar.
Class 10 · Maths · Chapter 6 · बिहार बोर्ड (BSEB)CBSE · NCERT 2026-27
Triangles
Triangles
How to use this page:
1. Read — Similar figures · basic proportionality · AAA/SSS/SAS · areas · Pythagoras, diagram, worked example, board tip
2. Check — each lesson has its own questions; the number follows the lesson
3. Mastery ★ — all of that lesson correct. Redo the wrong ones
4. The memory figure shows one triangle with a line parallel to the base and the intercept ratio AD/DB = AE/EC.
Progress stays in this browser.
🏅 Your achievements — 0/6 badges
🔥 0day streak · longest 0
⭐ 0total XP · today 0
🎓 1level
Next level: 150 XP to goright answer +10 · after a hint +5 · review +6 · badge +25
Last 7 days:
Everything is saved in this phone/browser — no login.
- 1 Similar figures
- 2 Basic proportionality — Thales
- 3 The converse of Thales
- 4 The criteria AAA, SSS and SAS
- 5 Areas of similar triangles
- 6 Pythagoras and the converse
- Chapter winner — every lesson at mastery ★
🎬 Shorts
This chapter has no Short on the channel yet, so ten empty cards are not shown. The notes, diagrams and checks are complete.
📄 PDF download — notes + answer key
Similar figures
समरूप आकृतियाँ · NCERT 6.2 · Exercise 6.1
Same shape, size not requiredNotes
Figures of the same shape are called similar. The size may differ. All circles are similar. All squares are similar. All equilateral triangles are similar.
Congruent figures have both the same shape and the same size. Every congruent pair is similar. Not every similar pair is congruent. A triangle and a square are not similar, because the shape is different. Exercise 6.1 is this recognition.
For a polygon, check both conditionsActivity
Two polygons with the same number of sides are similar only when corresponding angles are equal and corresponding sides are in the same ratio. Angles alone are not enough: a square and a long rectangle both have angles of 90°, but the sides are not proportional. A side ratio without the angles is not enough either.
Worked exampleExample
Question: Two squares have sides 4 cm and 6 cm. Are they similar? Are they congruent? Write the ratio.
Formula: The ratio of corresponding sides = smaller / larger.
Substitution: 4/6 = 2/3.
Answer: Both are squares, so they are similar. The size differs, so they are not congruent. The exact ratio is 2:3.
10-second revision
- Similar = same shape
- Congruent implies similar, not the reverse
- A polygon needs both equal angles and a side ratio
Board tip · BSEBBoard tip
In a BSEB answer write the sentence “similar but not congruent” with an example.
Board tip · CBSEBoard tip
On CBSE do not call a square and a rectangle similar just because the angles match.
Check your understandingall correct = mastery ★
2
Every similar pair is also congruent.
Explanation:
False — the size may differ.
3
The ratio of the sides of squares of 4 cm and 6 cm is ______.
💡 Hint:
4/6.
Explanation:
2:3.
4
Congruent triangles —
💡 Hint:
The size is equal too.
Explanation:
Congruent implies similar. The ratio is 1:1.
5
Why are two equilateral triangles similar? When will they be congruent?
Model answer (board format)
All three angles are 60°, so the shape is the same and they are similar. They are congruent only when the sides are equal, that is, the ratio is 1:1.
Check your answer:
Basic proportionality — Thales
आधारभूत समानुपातिकता — थेल्स · NCERT 6.3 · Exercise 6.2 · AD/DB = AE/EC
A parallel line cuts the sides in one ratioNotes
In triangle ABC, if DE is parallel to BC, with D on AB and E on AC, then AD/DB = AE/EC. Also AD/AB = AE/AC.
This is the basic proportionality theorem. Exercise 6.2 of the current NCERT is on it. The line must cut the sides at distinct points. Do not place D on B.
The parallel mark first, then the ratioActivity
Use the ratio only when the figure says DE || BC. AD and DB are pieces of one side. AE and EC are pieces of the other. Writing AD/AE = DB/EC scrambles the order, unless that form really is equivalent. Keep the unit centimetre.
Worked exampleExample
Question: DE || BC, AD = 3 cm, DB = 6 cm, AE = 4 cm. Find EC.
Formula: AD/DB = AE/EC.
Substitution: 3/6 = 4/EC. 1/2 = 4/EC. EC = 8.
Answer: EC = 8 cm. Check: AD/AB = 3/9 = 1/3 and AE/AC = 4/12 = 1/3.
10-second revision
- If DE || BC then AD/DB = AE/EC
- AD/AB = AE/AC is also correct
- Write the unit cm
Board tip · BSEBBoard tip
In a BSEB answer keep the formula line and the substitution apart. Do not jump to 8 cm.
Board tip · CBSEBoard tip
On CBSE write the form that is given. Do not swap AD/DB with AD/AB.
Check your understandingall correct = mastery ★
1
If AD = 3 cm, DB = 6 cm, AE = 4 cm and DE || BC, then EC is —
💡 Hint:
3/6 = 4/EC.
Explanation:
EC = 8 cm.
2
If DE || BC, then AD/AB = AE/AC.
Explanation:
True — this is the other form of Thales.
3
In Thales, AD/DB = AE/______.
💡 Hint:
The lower piece of the other side.
Explanation:
EC.
4
DE || BC, AD = 2 cm, DB = 3 cm, AE = 4 cm. Find EC and AC.
Model answer (board format)
AD/DB = AE/EC. 2/3 = 4/EC. EC = 6 cm. AC = AE + EC = 10 cm. Check: AD/AB = 2/5 and AE/AC = 4/10 = 2/5.
Check your answer:
The converse of Thales
थेल्स का विलोम · Equal ratios mean the line is parallel
Equal ratios prove the lines are parallelNotes
If a line divides two sides of a triangle in the same ratio, it is parallel to the third side. If AD/DB = AE/EC, then DE || BC.
This is the converse. The direct theorem assumes parallel and gives the ratio. The converse assumes the ratio and gives parallel. If the ratios differ, the line is not taken as parallel.
Make two fractions and compareActivity
AD = 2 cm, DB = 4 cm, AE = 3 cm, EC = 6 cm. 2/4 = 1/2 and 3/6 = 1/2. They are equal, so DE || BC. If EC were 5 cm, then 3/5 ≠ 1/2, and you would not call them parallel.
Worked exampleExample
Question: AD = 2 cm, DB = 4 cm, AE = 3 cm, EC = 6 cm. Is DE || BC?
Formula: Converse — if AD/DB = AE/EC then DE || BC.
Substitution: AD/DB = 2/4 = 1/2. AE/EC = 3/6 = 1/2.
Answer: Both ratios are 1/2, so DE is parallel to BC.
10-second revision
- Equal ratios give parallel
- Unequal ratios do not give parallel
- The direct theorem and the converse run in opposite directions
Board tip · BSEBBoard tip
In a BSEB answer write both fractions in lowest terms, then the equal sign.
Board tip · CBSEBoard tip
On CBSE, “almost equal” is not accepted. 1/2 and 3/5 are not equal.
Check your understandingall correct = mastery ★
1
If AD/DB = AE/EC = 1/2, then —
💡 Hint:
The converse.
Explanation:
DE is parallel to BC.
2
If AD/DB = 1/2 and AE/EC = 3/5, then DE || BC.
Explanation:
False — the ratios are not equal.
3
Both 2/4 and 3/6 equal ______.
💡 Hint:
Lowest terms.
Explanation:
1/2.
4
The direct Thales theorem starts from —
💡 Hint:
The converse starts from the ratio.
Explanation:
The direct theorem assumes parallel and gives the ratio.
5
In the converse, the conclusion is that the line is parallel.
Explanation:
True — the ratio is given, and parallel is to be proved.
6
AD = 4 cm, DB = 6 cm, AE = 6 cm, EC = 9 cm. What do you say about DE and BC?
Model answer (board format)
AD/DB = 4/6 = 2/3. AE/EC = 6/9 = 2/3. By the converse, DE || BC.
Check your answer:
The criteria AAA, SSS and SAS
कसौटी AAA, SSS और SAS · NCERT 6.4 · Exercise 6.3 · ΔABC ~ ΔDEF
The order of the letters is the correspondenceNotes
Two triangles are similar if corresponding angles are equal. The third angle is fixed by 180°, so AA is enough. This is called the AAA criterion.
SSS: the three corresponding sides are in one ratio. SAS: one angle is equal and both arms of that angle are in one ratio. ΔABC ~ ΔDEF means A↔D, B↔E, C↔F. The current exercise 6.3 is these criteria. SAS congruence asks for equal lengths; SAS similarity asks for a ratio.
Write the corresponding letters firstActivity
If the sides are 3, 4, 5 and 6, 8, 10, the ratio is 1:2. They are similar by SSS. They are not congruent, because the lengths are not equal. If both triangles have angles 50°, 60° and 70°, they are similar by AA, without measuring sides.
Worked exampleExample
Question: ΔABC ~ ΔPQR, AB = 4 cm, PQ = 6 cm. Which side corresponds to AB, and what is the ratio?
Formula: The order is A↔P, B↔Q, C↔R, so AB/PQ.
Substitution: AB/PQ = 4/6 = 2/3.
Answer: AB corresponds to PQ. The exact ratio is 2:3. Do not divide by QR.
10-second revision
- AA is enough for triangles
- SSS needs one ratio of all three sides
- The letter order of ~ is the correspondence
Board tip · BSEBBoard tip
In a BSEB answer write both the name of the criterion and the correspondence. Only the ~ symbol is incomplete.
Board tip · CBSEBoard tip
On CBSE do not write SAS similarity as SAS congruence. One asks for a ratio, the other for equal lengths.
Check your understandingall correct = mastery ★
1
Two triangles with angles 50°, 60°, 70° are similar by —
💡 Hint:
The angles are equal.
Explanation:
AAA, or AA.
2
Triangles with sides 3, 4, 5 and 6, 8, 10 are congruent.
Explanation:
False — the ratio is 1:2, so they are similar, not congruent.
3
In ΔABC ~ ΔPQR, the side corresponding to AB is ______.
💡 Hint:
The first two letters.
Explanation:
PQ.
4
SAS for similarity asks for —
💡 Hint:
The angle is included.
Explanation:
One equal angle and its arms in one ratio.
5
ΔABC ~ ΔDEF and AB = 5 cm, DE = 15 cm. Write the ratio and say which sides correspond.
Model answer (board format)
A↔D and B↔E, so DE corresponds to AB. AB/DE = 5/15 = 1/3. The exact ratio is 1:3.
Check your answer:
Areas of similar triangles
समरूप त्रिभुजों के क्षेत्रफल · Older Bihar exercise 6.4 · (side)²
The area ratio equals the square of the side ratioNotes
If ΔABC ~ ΔDEF, then ar(ABC)/ar(DEF) = (AB/DE)². The square is of the side, not of the area.
This is theorem 6.6 and exercise 6.4 of the older Bihar book. The current NCERT chapter 6 stops its exercises at the criteria in 6.3. Write area in square centimetres. If the side ratio is 2:3, the area ratio is 4:9.
The square root first, then the sideActivity
If the areas are 16 cm² and 36 cm², the ratio is 16/36 = 4/9. The side ratio is √(4/9) = 2/3. If the smaller corresponding side is 4 cm, the larger is 4 × 3/2 = 6 cm. Before the square root, see that the triangles are given as similar.
Worked exampleExample
Question: ΔABC ~ ΔDEF, the areas are 16 cm² and 36 cm². AB = 4 cm and AB corresponds to DE. Find DE.
Formula: ar(ABC)/ar(DEF) = (AB/DE)².
Substitution: 16/36 = (4/DE)². 4/9 = (4/DE)². 2/3 = 4/DE.
Answer: DE = 6 cm. Check: (4/6)² = 4/9 and 16/36 = 4/9.
10-second revision
- Area ratio = (side ratio)²
- A side ratio 2:3 gives an area ratio 4:9
- The unit is cm²
Board tip · BSEBBoard tip
In a BSEB answer write the square line on its own. Do not call 16/36 the side ratio.
Board tip · CBSEBoard tip
Current CBSE exercise 6.3 does not ask this result. Exercise 6.4 of the older book does.
Check your understandingall correct = mastery ★
1
If the side ratio is 2:3, the ratio of the areas is —
💡 Hint:
Take squares.
Explanation:
4:9.
2
The areas of similar triangles equal the ratio of the corresponding sides.
Explanation:
False — it is the ratio of the squares of the sides.
3
The side ratio for areas 16 cm² and 36 cm² is ______.
💡 Hint:
The square root of 4/9.
Explanation:
2:3.
4
If AB = 4 cm and the side ratio is 2:3, the larger corresponding side is —
💡 Hint:
4 × 3/2.
Explanation:
6 cm.
5
Similar triangles of equal area are congruent.
Explanation:
True — the side ratio is 1, so the size is equal too.
6
Corresponding sides of two similar triangles are 3 cm and 5 cm. The smaller area is 18 cm². Find the larger area.
Model answer (board format)
ar1/ar2 = (3/5)² = 9/25. 18/ar2 = 9/25. ar2 = 18 × 25/9 = 50. The larger area is 50 cm².
Check your answer:
Pythagoras and the converse
पाइथागोरस और उसका विलोम · Older Bihar exercise 6.5 · a² + b² = c²
The square of the hypotenuse is the sum of the other twoNotes
In a right triangle, hypotenuse² = base² + perpendicular². Converse: if the square of one side equals the sum of the squares of the other two, the angle opposite the longest side is a right angle.
Exercise 6.5 of the older Bihar book is this. If a perpendicular is drawn from the right-angled vertex to the hypotenuse, the two small triangles are similar to the whole triangle. The book reaches Pythagoras from that similarity. The current NCERT chapter 6 goes to the summary after the criteria.
Put the longest side in the hypotenuse placeActivity
Among 6 cm, 8 cm and 10 cm, the longest is 10 cm. Compare 6² + 8² with 10². If they match, the angle is a right angle. The same check works for 5, 12, 13: 25 + 144 = 169. The unit is centimetre on a side and square centimetre on a square.
Worked exampleExample
Question: The legs of a right triangle are 6 cm and 8 cm. Find the hypotenuse.
Formula: c² = a² + b².
Substitution: c² = 6² + 8² = 36 + 64 = 100.
Answer: c = 10 cm, because a length is positive. Check: 36 + 64 = 100.
10-second revision
- c² = a² + b²
- The converse gives a right angle opposite the longest side
- A negative length is not taken
Board tip · BSEBBoard tip
In exercise 6.5 of the older BSEB book, the line that adds the squares should be visible.
Board tip · CBSEBoard tip
The current CBSE exercises of chapter 6 do not ask Pythagoras separately. Write a similarity criterion there.
Check your understandingall correct = mastery ★
1
The hypotenuse of a right triangle with legs 6 cm and 8 cm is —
💡 Hint:
36 + 64 = 100.
Explanation:
10 cm.
2
A triangle of 5 cm, 12 cm and 13 cm is right-angled.
Explanation:
True — 25 + 144 = 169 = 13².
3
6² + 8² = ______.
💡 Hint:
36 + 64.
Explanation:
100. The hypotenuse is 10 cm.
4
Is a triangle of 7 cm, 24 cm and 25 cm right-angled? Check by the converse.
Model answer (board format)
7² + 24² = 49 + 576 = 625 = 25². By the converse, the angle opposite 25 cm is a right angle.
Check your answer:
❓ Full question bank — with answers and explanations — 64 questions
Pick a type. The 30 lesson checks are separate — each lesson has as many as its topic needs. All correct earns mastery ★.
BSEB modelpractice · not annual60 tried · 0 correctMultiple choiceMCQ120 tried · 0 correctTrue / falseTrue / false80 tried · 0 correctFill in the blanksFill80 tried · 0 correctMatchMatch20 tried · 0 correctAssertion–reasonAssertion–reason50 tried · 0 correctBalancingBalancing40 tried · 0 correctIdentifyIdentify20 tried · 0 correctShort and longVSA, SA, LA110 tried · 0 correctCBSE-stylenot a PYQ60 tried · 0 correct
No question is marked as a verified past paper. The BSEB set is a model for practice. CBSE items are CBSE-style, not a copy of any year’s paper.
Multiple choice
0/12Pick one option. A wrong try brings a hint.
1
These are always similar —
💡 Hint:
One shape.
Explanation:
All squares are similar.
2
If AD = 3 cm, DB = 6 cm, AE = 4 cm and DE || BC, EC is —
💡 Hint:
3/6 = 4/EC.
Explanation:
8 cm.
3
If AD/DB = AE/EC, then —
💡 Hint:
The converse.
Explanation:
DE is parallel to BC.
4
In ΔABC ~ ΔPQR, AB corresponds to —
💡 Hint:
The order of the letters.
Explanation:
PQ.
5
If the side ratio is 2:3, the area ratio is —
💡 Hint:
Squares.
Explanation:
4:9.
6
The hypotenuse for 6 cm and 8 cm is —
💡 Hint:
36 + 64.
Explanation:
10 cm.
7
Triangles 3, 4, 5 and 6, 8, 10 are —
💡 Hint:
The ratio is 1:2.
Explanation:
Similar by SSS, not congruent because the size differs. Both are also right-angled.
8
Squares of 4 cm and 6 cm are —
💡 Hint:
All squares are similar.
Explanation:
Similar, ratio 2:3. Not congruent.
9
If the areas are 16 cm² and 36 cm² and the smaller side is 4 cm, the larger corresponding side is —
💡 Hint:
2:3.
Explanation:
6 cm.
10
5, 12, 13 has a right angle because —
💡 Hint:
The sum of squares.
Explanation:
25 + 144 = 169 = 13².
11
This is not enough for polygons to be similar —
💡 Hint:
A rectangle and a square.
Explanation:
Angles alone are not enough. A side ratio is needed too.
12
AA is enough for a triangle because —
💡 Hint:
The sum of angles.
Explanation:
The third angle is fixed, so AAA and AA are the same statement.
True or false
0/81
All circles are similar.
Explanation:
True — the shape is the same.
2
Similar triangles are always congruent.
Explanation:
False — the size may differ.
3
If DE || BC, then AD/DB = AE/EC.
Explanation:
True — Thales.
4
If the ratios are unequal, the converse still makes the line parallel.
Explanation:
False — the ratios must be equal.
5
In ΔABC ~ ΔDEF, EF corresponds to AB.
Explanation:
False — DE corresponds to AB.
6
The ratio of the areas of similar triangles equals the square of the ratio of corresponding sides.
Explanation:
True.
7
6² + 8² = 10².
Explanation:
True — 36 + 64 = 100.
8
A square and a long rectangle are similar because the angles are equal.
Explanation:
False — the sides are not proportional.
Fill in the blanks
0/81
If DE || BC, then AD/DB = AE/______.
💡 Hint:
EC.
Explanation:
The lower piece.
2
The ratio of 4 cm and 6 cm is ______.
💡 Hint:
2:3.
Explanation:
4/6.
3
If the sides are 2:3, the areas are ______.
💡 Hint:
4:9.
Explanation:
Squares.
4
6² + 8² = ______.
💡 Hint:
100.
Explanation:
The square of the hypotenuse.
5
In ΔABC ~ ΔDEF, the angle corresponding to angle B is ______.
💡 Hint:
Angle E.
Explanation:
The second letter.
6
The positive square root of 16/36 is ______.
💡 Hint:
2/3.
Explanation:
4/6 in lowest terms.
7
5² + 12² = ______.
💡 Hint:
169.
Explanation:
13².
8
The side ratio of congruent figures is ______.
💡 Hint:
1:1.
Explanation:
Equal size.
Match
0/21
Match the name with its meaning.
Column B: A. a² + b² = c² · B. AD/DB = AE/EC · C. Equal angles · D. (side)²
1. Thales
2. AAA
3. Areas
4. Pythagoras
Explanation:
Thales is the ratio, AAA is the angles, areas use the square, and Pythagoras is the square of the hypotenuse.
2
Match the exercise with its job.
Column B: A. Pythagoras · B. Similar figures · C. The Thales ratio · D. AAA, SSS, SAS
1. 6.1
2. 6.2
3. 6.3
4. 6.5 of the older book
Explanation:
6.1 is similar figures, 6.2 is Thales, 6.3 is the criteria. Pythagoras is 6.5 of the older book.
Assertion–reason
0/5Check both statements, then see whether the reason explains the assertion.
1
Assertion (A): If DE || BC and AD = 3 cm, DB = 6 cm, AE = 4 cm, then EC = 8 cm.
Reason (R): AD/DB = AE/EC.
Explanation:
Both are true and R is the basis of the calculation.
2
Assertion (A): All circles are similar.
Reason (R): The ratio of the areas of similar triangles equals the square of the sides.
Explanation:
Both are true, but R does not explain why circles are similar.
3
Assertion (A): Triangles with sides 3, 4, 5 and 6, 8, 10 are similar.
Reason (R): They are congruent as well.
Explanation:
A is true, by SSS and the ratio 1:2. R is false.
4
Assertion (A): 6 cm, 8 cm and 11 cm are the sides of a right triangle.
Reason (R): In the converse, the square of the longest side must equal the sum of the squares of the other two.
Explanation:
A is false, 36 + 64 = 100 ≠ 121. R is true.
5
Assertion (A): Squares of 4 cm and 6 cm are similar but not congruent.
Reason (R): The shape is the same and the size ratio is 2:3, not 1:1.
Explanation:
Both are true and R explains A.
Coefficient practice
0/4These chemistry lines are not a result of this maths chapter. They are only practice in filling coefficients. A blank means 1.
1
This is coefficient practice and not a result of this maths chapter. In the line where H2 + O2 makes water, Fill coefficients (blank = 1).
H2 + O2 → H2O
Explanation: 2H2 + O2 → 2H2O
2
This is coefficient practice and not a result of this maths chapter. In the line where N2 + H2 makes ammonia, Fill coefficients (blank = 1).
N2 + H2 → NH3
Explanation: N2 + 3H2 → 2NH3
3
This is coefficient practice and not a result of this maths chapter. In the line where C + O2 makes carbon dioxide, Fill coefficients (blank = 1).
C + O2 → CO2
Explanation: C + O2 → CO2
4
This is coefficient practice and not a result of this maths chapter. In the burning line of CH4 + O2, Fill coefficients (blank = 1).
CH4 + O2 → CO2 + H2O
Explanation: CH4 + 2O2 → CO2 + 2H2O
Classify
0/21
Place each pair as similar, congruent, or neither.
Squares of 4 cm and 6 cm
Two circles of equal size
A triangle and a square
Sides 3, 4, 5 and 6, 8, 10
Explanation:
The squares are similar. Equal circles are also congruent. A triangle and a square have different shapes. 3-4-5 and 6-8-10 are similar.
2
Place each criterion with its mark.
All three corresponding angles equal
All three sides in one ratio
One angle and the ratio of its arms
Hypotenuse² = sum of the squares of the other two
Explanation:
Angles are AAA, three sides are SSS, the included angle is SAS, and the hypotenuse is Pythagoras.
Very short answer
0/41
Define similar figures in one line.
Model answer (board format)
Figures of the same shape are similar; the size need not be equal.
Check your answer:
2
Write the Thales ratio.
Model answer (board format)
If DE || BC, then AD/DB = AE/EC.
Check your answer:
3
Write the area formula for similar triangles.
Model answer (board format)
ar(ABC)/ar(DEF) = (AB/DE)² when ΔABC ~ ΔDEF.
Check your answer:
4
Write Pythagoras’ theorem.
Model answer (board format)
In a right triangle, hypotenuse² = the sum of the squares of the other two sides.
Check your answer:
Short answer
0/41
DE || BC, AD = 5 cm, DB = 10 cm, EC = 8 cm. Find AE.
Model answer (board format)
AD/DB = AE/EC. 5/10 = AE/8. 1/2 = AE/8. AE = 4 cm.
Check your answer:
2
ΔABC ~ ΔDEF, AB = 6 cm, DE = 9 cm, BC = 8 cm. Find EF.
Model answer (board format)
AB/DE = BC/EF. 6/9 = 8/EF. 2/3 = 8/EF. EF = 12 cm.
Check your answer:
3
Corresponding sides of similar triangles are 4 cm and 6 cm. The smaller area is 24 cm². Find the larger area.
Model answer (board format)
(4/6)² = 16/36 = 4/9. 24/ar = 4/9. ar = 24 × 9/4 = 54. The larger area is 54 cm².
Check your answer:
4
Find the hypotenuse of the right triangle with legs 9 cm and 12 cm.
Model answer (board format)
c² = 81 + 144 = 225. c = 15 cm.
Check your answer:
Long answer
0/31
In triangle ABC, DE || BC, AD = 4 cm, DB = 4 cm, AE = 5 cm. Find EC. Then say what AD/AB and AE/AC give.
Model answer (board format)
AD/DB = AE/EC. 4/4 = 5/EC. EC = 5 cm. AB = 8 cm, AC = 10 cm. AD/AB = 4/8 = 1/2 and AE/AC = 5/10 = 1/2. Both forms give the same ratio.
Check your answer:
2
Two triangles have sides 5, 12, 13 cm and 10, 24, 26 cm. Write the similarity criterion, the ratio, and whether they are right-angled. Also find the ratio of the areas.
Model answer (board format)
10/5 = 24/12 = 26/13 = 2. Similar by SSS, ratio 1:2. 25 + 144 = 169 = 13², so the first is right-angled, and so is the second, because 100 + 576 = 676 = 26². The area ratio is (1/2)² = 1:4.
Check your answer:
3
A pole casts a 6 m shadow, and at the same time a 2 m peg casts a 1.5 m shadow. Find the height of the pole. Which similarity is used?
Model answer (board format)
The sun makes the same angle, so the triangles are similar by AA. Height over shadow is equal: h/6 = 2/1.5. h = 6 × 2/1.5 = 6 × 4/3 = 8. The pole is 8 m high.
Check your answer:
BSEB model paper · practice
0/6This model set is for practice. It is not a question from any year’s annual examination.
1
Which of these is a congruent pair?
💡 Hint:
The size is the same too.
Explanation:
Circles of equal radius are congruent.
2
If AD = 2 cm, DB = 3 cm, AE = 4 cm and DE || BC, EC is —
💡 Hint:
2/3 = 4/EC.
Explanation:
EC = 6 cm.
3
9² + 12² = ______.
💡 Hint:
225.
Explanation:
The hypotenuse is 15 cm.
4
AD = 3 cm, DB = 5 cm, AE = 4 cm, EC = 6 cm. Is DE || BC? Show the ratios.
Model answer (board format)
AD/DB = 3/5. AE/EC = 4/6 = 2/3. 3/5 ≠ 2/3, so DE is not parallel to BC.
Check your answer:
5
A right triangle has perpendicular 8 cm and hypotenuse 17 cm. Find the base.
Model answer (board format)
17² = 8² + b². 289 = 64 + b². b² = 225. b = 15 cm.
Check your answer:
6
Assertion (A): All equilateral triangles are similar.
Reason (R): Their sides are always equal, so they are congruent.
Explanation:
A is true, the angles are 60°. R is false, the sides may differ.
CBSE-style questions
0/6These are competency-based practice questions. They are not a copy of any year’s paper.
1
A student writes AB/EF in ΔABC ~ ΔDEF. AB = 4 cm, DE = 8 cm, EF = 10 cm. The correct corresponding ratio is —
💡 Hint:
Letter order A-D, B-E.
Explanation:
DE corresponds to AB. 4/8 = 1/2. EF does not correspond.
2
A wall casts a 9 m shadow. At the same time a 1.5 m stick casts a 1 m shadow. The height of the wall is —
💡 Hint:
h/9 = 1.5/1.
Explanation:
h = 9 × 1.5 = 13.5 m. AA similarity.
3
Assertion (A): Similar triangles of areas 16 cm² and 36 cm² have corresponding sides in the ratio 2:3.
Reason (R): The ratio of the areas is the square of the ratio of the sides.
Explanation:
Both are true. 16/36 = 4/9 = (2/3)².
4
Assertion (A): Any pair of rectangles is similar.
Reason (R): For polygons, a ratio of corresponding sides is needed along with the corresponding angles.
Explanation:
A is false. A square and a long rectangle are not similar. R is true.
5
A ladder rests on the ground 8 m from a wall and reaches 15 m up the wall. Find the length of the ladder.
Model answer (board format)
It is a right angle. c² = 8² + 15² = 64 + 225 = 289. c = 17 m.
Check your answer:
6
What does exercise 6.3 of the current book ask? In the older book, at which numbers are areas and Pythagoras?
Model answer (board format)
Exercise 6.3 of the current NCERT asks the similarity criteria AAA, SSS and SAS. Areas are exercise 6.4 of the older Bihar book, and Pythagoras is 6.5.
Check your answer:
🏛️ Board exam corner — Bihar Board (BSEB)— CBSE
Switch board with BSEB | CBSE above. The lessons follow the same NCERT chapter.
BSEB model · practiceBoard tip
This page has no verified annual-exam question, because no source page has been added. The model set below is practice in the board pattern.
🏛️ Model questions on one page →
The verified label will be used only when a source page for the question is available.
CBSE-style · competency-based
These are case and assertion-reason practice items. Do not treat them as past CBSE questions.
🔁 Spaced review — today’s questions
Wrong questions return soon; correct ones return after a few days.
🧠 What you learned + equation sheet
What you learned
- Similar means the same shape; the size need not be equal.
- Congruent figures are similar, but not every similar pair is congruent.
- If DE || BC, then AD/DB = AE/EC. That is basic proportionality.
- For triangles, AAA, SSS or SAS is enough for similarity. Letter order is the correspondence.
- The ratio of the areas of similar triangles equals the square of the ratio of corresponding sides.
- In a right triangle, hypotenuse² = base² + perpendicular². The converse is also true.
| What | Keep this |
|---|---|
| Similar | same shape, size may differ |
| Congruent | same shape and same size |
| Thales | AD/DB = AE/EC if DE || BC |
| AAA / AA | equal corresponding angles |
| Areas | ar ratio = (side ratio)² |
| Pythagoras | a² + b² = c² |
The notes are original writing. The textbook was used only for activity order and numbers. “Verified” will be used only on a question that has a source page.