Chapter 6: Triangles

Important questions — the ones the board asked most (20)

1. Q1Long answer5 marksAsked: 2026

   (A) State and prove Basic Proportionality Theorem. OR (B) In the given figure, CM and RN are the medians of triangle ABC and triangle PQR respectively. If triangle ABC ~ triangle PQR, then prove that : (i) triangle AMC ~ triangle PNR (ii) triangle CMB ~ triangle RNQ

   

   Medha Ma’am

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   Model answer · Basic Proportionality Theorem proof

   1. 1.Basic Proportionality Theorem (Thales Theorem): If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.1 mark
   2. 2.Given: A triangle ABC in which a line parallel to side BC intersects the other two sides AB and AC at D and E respectively, so DE || BC. To prove: AD/DB = AE/EC. Construction: Join BE and CD. Draw DM ⊥ AC, with M on AC, and EN ⊥ AB, with N on AB. Figure: Triangle ABC with D on AB, E on AC, DE || BC, the segments BE and CD joined, and the perpendiculars DM and EN drawn.2 marks
   3. 3.Proof: Area of a triangle = 1/2 × base × height. ar(ADE) = 1/2 × AD × EN and ar(BDE) = 1/2 × DB × EN (EN is the height of both triangles). So, ar(ADE)/ar(BDE) = AD/DB ... (1) Similarly, ar(ADE) = 1/2 × AE × DM and ar(DEC) = 1/2 × EC × DM (DM is the height of both triangles). So, ar(ADE)/ar(DEC) = AE/EC ... (2) Triangles BDE and DEC lie on the same base DE and between the same parallels BC and DE. So, ar(BDE) = ar(DEC) ... (3) From (1), (2) and (3): AD/DB = AE/EC. Hence proved.2 marks

   Diagram: Draw triangle ABC with point D on AB and point E on AC so that DE || BC. Join BE and CD, and draw DM ⊥ AC and EN ⊥ AB. Label A, B, C, D, E, M and N.

   • Write all four parts with their headings: Given, To prove, Construction, and a labelled figure that shows DE || BC.
   • Give the reason for ar(BDE) = ar(DEC): the two triangles are "on the same base DE and between the same parallels BC and DE".

   Marks split as in the board’s published marking scheme

2. Q2Long answer5 marksAsked: 2026

   (a) Prove that if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio. OR (b) As shown in the given figure, a girl of height 90 cm is walking away from the base of a lamp post at a speed of 1.2 m/s. If the lamp is 3.6 m above the ground, find the length of her shadow after 4 seconds.

   

   Medha Ma’am

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   Hear it from Medha Ma’am · 135 s

   Model answer · Basic Proportionality Theorem proof

   1. 1.Given: A triangle ABC in which a line parallel to side BC intersects the other two sides AB and AC at D and E respectively, so DE || BC. To prove: AD/DB = AE/EC. Construction: Join BE and CD, and then draw DM ⊥ AC and EN ⊥ AB. Figure: Draw triangle ABC with D on AB and E on AC so that DE || BC. Join BE and CD. Draw the perpendicular DM from D to AC and the perpendicular EN from E to AB.2 marks
   2. 2.Proof: ar(ADE) = 1/2 × AD × EN (area of a triangle = 1/2 base × height) ar(BDE) = 1/2 × DB × EN (same height EN) So, ar(ADE)/ar(BDE) = (1/2 × AD × EN)/(1/2 × DB × EN) = AD/DB ...(1) ar(ADE) = 1/2 × AE × DM (taking AE as the base and DM as the height) ar(DEC) = 1/2 × EC × DM (same height DM) So, ar(ADE)/ar(DEC) = (1/2 × AE × DM)/(1/2 × EC × DM) = AE/EC ...(2) ΔBDE and ΔDEC are on the same base DE and between the same parallels BC and DE. So, ar(BDE) = ar(DEC) ...(3) From (1), (2) and (3): AD/DB = AE/EC. So the line parallel to BC divides the other two sides in the same ratio. Hence proved.3 marks

   Diagram: Draw triangle ABC with a line DE parallel to BC, D on AB and E on AC. Join BE and CD, and draw DM perpendicular to AC and EN perpendicular to AB. Label A, B, C, D, E, M and N.

   • Write the construction in full: "Join BE and CD, and then draw DM ⊥ AC and EN ⊥ AB."
   • Give the reason for ar(BDE) = ar(DEC): "on the same base DE and between the same parallels BC and DE".
   • Number the area ratios (1), (2) and (3), then combine them to get AD/DB = AE/EC.

   Marks split as in the board’s published marking scheme

3. Q3Long answer5 marksAsked: 2026

   In triangle ABC, AD is a median. X is a point on AD such that AX : XD = 2 : 3. BX is extended so that it intersects AC at Y. Prove that BX = 4 XY.

   

   Medha Ma’am

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   Hear it from Medha Ma’am · 155 s

   Model answer · Basic Proportionality Theorem, proof

   1. 1.Given: In ΔABC, AD is a median, so D is the mid-point of BC. X is on AD with AX : XD = 2 : 3, and BX extended meets AC at Y. To prove: BX = 4 XY. Construction: Through D, draw DZ || BY, meeting AC at Z.0.5 marks
   2. 2.In ΔCBY, DZ || BY. By the Basic Proportionality Theorem, CD/DB = CZ/ZY. Since D is the mid-point of BC, CD = DB, so CD/DB = CZ/ZY = 1. So Z is the mid-point of CY.1 mark
   3. 3.In ΔCDZ and ΔCBY, ∠C is common and ∠CDZ = ∠CBY (corresponding angles, as DZ || BY), so ΔCDZ ~ ΔCBY and DZ/BY = CD/CB = 1/2. Therefore, DZ = 1/2 BY --- (i)0.5 marks
   4. 4.In ΔADZ, DZ || XY, because XY lies along BY. ∠A is common and ∠AXY = ∠ADZ (corresponding angles). So, ΔAXY ~ ΔADZ, and AX/AD = XY/DZ.1 mark
   5. 5.AX : XD = 2 : 3, so AD = AX + XD = 5 parts and AX/AD = 2/5. 2/5 = XY/DZ or DZ = 5/2 XY ---- (ii)0.5 marks
   6. 6.Using (i) and (ii): 1/2 BY = 5/2 XY BY = 5 XY1 mark
   7. 7.Therefore BX = BY − XY = 5 XY − XY = 4 XY. Hence proved.0.5 marks

   Diagram: Draw triangle ABC with D the mid-point of BC and AD joined. Mark X on AD, extend BX to meet AC at Y, then draw DZ parallel to BY meeting AC at Z. Label A, B, C, D, X, Y, Z.

   • Begin with the construction "Through D, draw DZ || BY, meeting AC at Z" and draw it in the figure.
   • Write "CD/DB = CZ/ZY = 1" and give the reason: the Basic Proportionality Theorem plus D being the mid-point of BC.
   • Number the results (i) and (ii), then finish with "BX = BY − XY = 4 XY".

   Marks split as in the board’s published marking scheme

4. Q4Long answer5 marksAsked: 2026

   In Δ ABC, AD is a median. X is a point on AD such that AX : XD = 2 : 3. BX is extended so that it intersects AC at Y. Prove that BX = 4 XY.

5. Q5Long answer5 marksAsked: 2026

   33. (a) If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then prove that the other two sides are divided in the same ratio. OR (b) AD and PS are respectively, the medians of delta ABC and delta PQR. If delta ABC ~ delta PQR, then prove that (i) delta ADC ~ delta PSR (ii) AD/PS = BC/QR

   How the model answer begins

   1. Given: In triangle ABC, a line DE is drawn parallel to side BC (DE || BC) and it intersects the other two sides AB and AC at the distinct points D and E respectively. To Prove: AD/DB = AE/EC. Construction: Join BE and…

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6. Q6Long answer5 marksAsked: 2026

   (a) If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then prove that the other two sides are divided in the same ratio. OR (b) AD and PS are respectively, the medians of triangle ABC and triangle PQR. If triangle ABC ~ triangle PQR, then prove that (i) triangle ADC ~ triangle PSR (ii) AD/PS = BC/QR

   How the model answer begins

   1. Given: A triangle ABC in which a line parallel to side BC intersects the other two sides AB and AC at distinct points D and E respectively, i.e., DE || BC. To prove: AD/DB = AE/EC. Construction: Join BE and CD, and then…

   The full answer from Medha Ma’am →
7. Q7Long answer5 marksAsked: 2025

   State the SAS criteria of similarity of two triangles. In the given figure, it is given that OA . OC = OB . OD. Use the SAS criteria to prove that AD || CB.

   How the model answer begins

   1. SAS (Side–Angle–Side) similarity criterion: If one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then the two triangles are similar.

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8. Q8Long answer5 marksAsked: 2025

   State AA criterion of similarity of two triangles and use it to prove the following. In the given figures of delta ABC and delta PQR, AD and PS are angle bisectors of angle BAC and angle RPQ respectively. If delta ABC ~ delta PQR, prove that delta ACD ~ delta PRS.

   How the model answer begins

   1. AA similarity criterion: If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar.

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9. Q9Long answer5 marksAsked: 2025

   34. State and prove "Basic Proportionality Theorem."

   How the model answer begins

   1. Statement (Basic Proportionality Theorem): If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.

   The full answer from Medha Ma’am →
10. Q10Long answer5 marksAsked: 2025

    (a) State and Prove "Basic Proportionality Theorem". OR (b) In the given figure, CM and RN are respectively, the medians of triangle ABC and triangle PQR. If triangle ABC ~ triangle PQR, prove that : (i) triangle AMC ~ triangle PNR (ii) angle BCM = angle QRN (iii) triangle BMC ~ triangle QNR

    How the model answer begins

    1. Basic Proportionality Theorem (Thales Theorem): If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.

    The full answer from Medha Ma’am →
11. Q11Long answer5 marksAsked: 2025

    (A) If a line is drawn parallel to one side of a triangle to intersect the other two sides at distinct points, prove that the other two sides are divided in the same ratio. Or (B) In delta ABC, points P and Q are situated on sides AB and AC respectively such that PQ || BC. Prove that the median AD drawn from point A to side BC bisects the line segment PQ.

    How the model answer begins

    1. Figure: Draw triangle ABC with a line parallel to side BC that meets AB at D and AC at E. Join BE and CD. Draw DM ⊥ AC and EN ⊥ AB.

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12. Q12Long answer5 marksAsked: 2025

    (a) If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then prove that the other two sides are divided in the same ratio. OR (b) Given that the sides AB and AC and median AD of a triangle ABC are respectively proportional to the sides PQ and PR and median PM of another triangle PQR. Show that triangle ABC ~ triangle PQR.

    How the model answer begins

    1. Figure: Triangle ABC with a line parallel to BC that meets AB at D and AC at E. BE and DC are joined. DG is drawn perpendicular to AC and EF is drawn perpendicular to AB.

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13. Q13Long answer5 marksAsked: 2025

    35. (a) If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then prove that the other two sides are divided in the same ratio. OR (b) It is given that sides AB and AC and median AD of delta ABC are respectively proportional to sides PQ and PR and median PM of another delta PQR. Show that delta ABC ~ delta PQR.

    How the model answer begins

    1. (a) Given: in ΔABC, DE ∥ BC with D on AB and E on AC. To prove: AD/DB = AE/EC. Construction: join BE and CD; draw DM ⊥ AC and EN ⊥ AB. OR (b) Given: AB/PQ = AC/PR = AD/PM, AD and PM medians. Construction: produce AD to…

    The full answer from Medha Ma’am →
14. Q14Long answer5 marksAsked: 2024

    (A) If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then prove that the other two sides are divided in the same ratio. OR (B) In the given figure PA, QB and RC are each perpendicular to AC. If AP = x, BQ = y and CR = z, then prove that 1/x + 1/z = 1/y

    How the model answer begins

    1. Attempting (A) – Basic Proportionality Theorem. Given: A triangle ABC in which a line parallel to side BC intersects the other two sides AB and AC at D and E respectively, i.e., DE || BC. To prove: AD/DB = AE/EC.…

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15. Q15Long answer5 marksAsked: 2024

    (a) If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then prove that the other two sides are divided in the same ratio. OR (b) Sides AB and AC and median AD to triangle ABC are respectively proportional to sides PQ and PR and median PM of another triangle PQR. Show that triangle ABC ~ triangle PQR.

    How the model answer begins

    1. Answering (a). Given: A triangle ABC in which a line parallel to side BC intersects the other two sides AB and AC at distinct points D and E respectively, that is, DE || BC. To prove: AD/DB = AE/EC. Construction: Join…

    The full answer from Medha Ma’am →
16. Q16Long answer5 marksAsked: 2024

    (a) In the given figure, Delta FEC is congruent to Delta GDB and angle 1 = angle 2. Prove that Delta ADE is similar to Delta ABC. OR (b) The sides AB and AC and median AD of a Delta ABC are respectively proportional to the sides PQ and PR and median PM of another triangle Delta PQR. Show that Delta ABC is similar to Delta PQR.

    How the model answer begins

    1. Given: in triangle ABC, D is a point on AB, E is a point on AC, and F and G are points on BC. Let ∠1 = ∠ADE, ∠2 = ∠AED, ∠3 = ∠ABC and ∠4 = ∠ACB. We are given ΔFEC ≅ ΔGDB, with F ↔ G, E ↔ D and C ↔ B, and ∠1 = ∠2. To…

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17. Q17Long answer5 marksAsked: 2024

    In the given figure, Delta FEC = Delta GDB and angle 1 = angle 2. Prove that Delta ADE ~ Delta ABC. OR Sides AB and AC and median AD of a Delta ABC are respectively proportional to sides PQ and PR and median PM of another Delta PQR. Show that Delta ABC ~ Delta PQR.

    How the model answer begins

    1. Given: In ΔABC and ΔPQR, AD and PM are medians and AB/PQ = AC/PR = AD/PM. To prove: ΔABC ~ ΔPQR. Figure: ΔABC with median AD, where D is the mid-point of BC, and ΔPQR with median PM, where M is the mid-point of QR. Let…

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18. Q18Long answer5 marksAsked: 2024

    (a) In the given figure, altitudes CE and AD of triangle ABC intersect each other at the point P. Show that (i) triangle AEP ~ triangle CDP (ii) triangle ABD ~ triangle CBE (iii) triangle AEP ~ triangle ADB OR (b) AD and PM are medians of triangles ABC and PQR, respectively, where triangle ABC ~ triangle PQR. Prove that AB/PQ = AD/PM.

    How the model answer begins

    1. Given: CE and AD are altitudes of triangle ABC, so CE ⊥ AB and AD ⊥ BC, and they intersect at P. (i) In triangle AEP and triangle CDP: ∠AEP = ∠CDP (each 90°, as CE ⊥ AB and AD ⊥ BC) ∠APE = ∠CPD (vertically opposite…

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19. Q19Long answer5 marksAsked: 2024

    (a) In the given figure, altitudes CE and AD of delta ABC intersect each other at the point P. Show that (i) delta AEP ~ delta CDP (ii) delta ABD ~ delta CBE (iii) delta AEP ~ delta ADB OR (b) AD and PM are medians of triangles ABC and PQR, respectively, where delta ABC ~ delta PQR. Prove that AB/PQ = AD/PM.

    How the model answer begins

    1. Given: AD and PM are medians of triangles ABC and PQR, respectively, and triangle ABC ~ triangle PQR. To prove: AB/PQ = AD/PM. Proof: triangle ABC ~ triangle PQR (Given), so their corresponding sides are in the same…

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20. Q20Long answer5 marksAsked: 2024

    (a) If BD and QM are the medians of triangles ABC and PQR respectively, where triangle ABC ~ triangle PQR, then prove that AB/PQ = BD/QM. OR (b) CD and GH are respectively the bisectors of angle ACB and angle EGF such that D and H lie on sides AB and FE of triangle ABC and triangle FEG respectively. If triangle ABC ~ triangle FEG, then show that: (i) CD/GH = AC/FG (ii) triangle DCB ~ triangle HGE

    How the model answer begins

    1. Given: ΔABC ~ ΔPQR. BD is the median of ΔABC, so D is the mid-point of AC. QM is the median of ΔPQR, so M is the mid-point of PR. To prove: AB/PQ = BD/QM. Figure: draw ΔABC with median BD to side AC, and ΔPQR with…

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113 more questions of this chapter, each with its model answer, in the Ask Medha app.

Objective questions with the board’s key (10)

1. 1. CBSE 2026

   If triangle ABC and triangle DEF are similar, where 2 AB = DE and BC = 8 cm, then EF is equal to :

   • (A) 4 cm
   • (B) 8 cm
   • (C) 12 cm
   • (D) 16 cm ✓

   Answer: (D)

2. 2. CBSE 2026

   Three triangles are given. For each triangle, the measure of two adjacent sides and the angle between them is given : Which of these triangles are similar?

   • (A) triangle RPQ and triangle XZY ✓
   • (B) triangle RPQ and triangle MNL
   • (C) triangle XZY and triangle MNL
   • (D) triangle RPQ, triangle XZY and triangle MNL are similar to each other.

   Answer: (A)

3. 3. CBSE 2026

   In triangles ABC and PQR, angle A = angle Q and angle B = angle R, then AB : AC is equal to :

   • (A) PQ : PR
   • (B) PQ : QR
   • (C) QR : QP ✓
   • (D) PR : QR

   Answer: (C)

4. 4. CBSE 2026

   In the given figure triangle ABC is shown, in which DE || BC. If AD = 5 cm, DB = 2.5 cm and DE = 8 cm, then the length of BC is :

   • (A) 10 cm
   • (B) 6 cm
   • (C) 12 cm ✓
   • (D) 7.5 cm

   Answer: (C)

5. 5. CBSE 2026

   In triangle DEF, AB || EF. The value of x is :

   • (A) 0, 2
   • (B) only 2 ✓
   • (C) - 2
   • (D) 1

   Answer: (B)

6. 6. CBSE 2026

   It is given that triangle ABC ~ triangle QRP such that AB = 9 cm, BC = 5 cm and PR = 2 cm. Length of side QR is :

   • (A) 0.9 cm
   • (B) 5/18 cm
   • (C) 10/9 cm
   • (D) 3.6 cm ✓

   Answer: (D)

7. 7. CBSE 2026

   Given that Delta ABC ~ Delta QRP where AB = 9 cm, BC = 5 cm and PR = 2 cm. The length of side QR is :

   • (A) 0.9 cm
   • (B) 5/18 cm
   • (C) 10/9 cm
   • (D) 3.6 cm ✓

   Answer: (D)

8. 8. CBSE 2026

   Given that triangle ABC ~ triangle QRP where AB = 9 cm, BC = 5 cm and PR = 2 cm. The length of side QR is :

   • (A) 0.9 cm
   • (B) 5/18 cm
   • (C) 10/9 cm
   • (D) 3.6 cm ✓

   Answer: (D)

9. 9. CBSE 2026

   In the given figure, OA * OB = OC * OD. Which of the following option is correct ?

   • (A) angle A = angle C ✓
   • (B) angle A = angle B
   • (C) angle A = angle D
   • (D) triangle OAD ~ triangle OBC

   Answer: (A)

10. 10. CBSE 2026

    In the given figure, PQ || BC. If AP : AB = 3 : 7, then AQ : QC is equal to :

    • (A) 3 : 7
    • (B) 3 : 10
    • (C) 7 : 3
    • (D) 3 : 4 ✓

    Answer: (D)

Questions parents and students ask

How many marks does "Triangles" carry in CBSE Class 10 Maths?

In the CBSE papers of 2023–2026, this chapter carried about 8 marks and about 3 questions a year; by marks it ranks 1 in Maths.

Which are the most important (VVI) questions of "Triangles"?

The 20 questions above are from the board’s real papers, each with the years it was asked; the most-asked come first with their full model answers. All 133 written questions of this chapter, with answers, are in the Ask Medha app.

Do the answers follow the board’s marking scheme?

Yes. CBSE publishes its marking schemes; every model answer here is written point by point to the marks the board’s own scheme gives, and checked by a second, independent model.