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CBSE · Class 10 · Mathematics

Triangles 6.3: Similarity of Triangles (Basic Proportionality Theorem)

5 concepts · 40 practice questions with answers · Free to read

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Applying the Basic Proportionality Theorem

From daily life

A shelf parallel to the floor in a triangular roof cuts both slanting sides in the same proportion.

Simple definition

The Basic Proportionality Theorem (Thales): if a line is drawn parallel to one side of a triangle, it divides the other two sides in the same ratio. In ΔABC with DE ∥ BC: AD/DB = AE/EC.

Formula: AD/DB = AE/EC

ABCDE462x

In the figure: DE ∥ BC: 4/6 = 2/x, so x = 3.

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Applying the Basic Proportionality Theorem

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Rule 1: Find the missing part — Use upper part / lower part on both sides.

  • Easy: AD = 4, DB = 6, AE = 2: EC = 3.
  • Medium: AD = 3, DB = 5, AE = 6: EC = 10.
  • Exam-type: AD = 2, DB = 3, EC = 6: AE = 4.

Rule 2: Other triangles — The names change, the rule does not: parallel line, same ratio.

  • Easy: ΔPQR, ST ∥ QR: PS = 3, SQ = 2, PT = 6 gives TR = 4.
  • Medium: ΔXYZ, MN ∥ YZ: XM = 5, MY = 10, XN = 4 gives NZ = 8.
  • Exam-type: ΔABC, DE ∥ BC: AD = 2, AB = 8, AE = 3 gives EC = 9.
Remember

Always parts of the same side compared with parts of the other side.

Common mistake

Comparing AD/DB with EC/AE, which is upside down.

Memory clue

"Parallel line cuts both sides in the same ratio."

Practice questions

Easy
  1. 1In △MLC, WT ∥ LC with W on ML and T on MC. If WL = 6 cm, MT = 8 cm, TC = 4 cm, find MW (in cm).
  2. 2In △EZT, PB ∥ ZT with P on EZ and B on ET. If EP = 5 cm, PZ = 10 cm, BT = 10 cm, find EB (in cm).
  3. 3In △AHF, TD ∥ HF with T on AH and D on AF. If AT = 8 cm, TH = 6 cm, DF = 9 cm, find AD (in cm).
Medium
  1. 4In △PQX, TF ∥ QX with T on PQ and F on PX. If PT = 3 cm, TQ = 5 cm, PF = 15 cm, find FX (in cm).
  2. 5In △CWT, PG ∥ WT with P on CW and G on CT. If CP = 18 cm, PW = 15 cm, GT = 15 cm, find CG (in cm).
  3. 6In △ZCD, YG ∥ CD. If ZG = 3 cm, GD = 4 cm, ZY = 6 cm, find the full length ZC (in cm).
Challenging
  1. 7In △SZQ, PA ∥ ZQ. If SA = 6 cm, AQ = 7 cm, SP = 24 cm, find the full length SZ (in cm).
  2. 8In △KDE, TB ∥ DE. If KT = 15 cm, TD = 30 cm, KB = 3 cm, find the full length KE (in cm).
Show answers (try them yourself first)
  1. 112
  2. 25
  3. 312
  4. 425
  5. 518
  6. 614
  7. 752
  8. 89

Converse: is the line parallel?

From daily life

To check that a shelf is level, measure the parts it cuts on both sides; equal ratios prove it is parallel.

Simple definition

The converse: 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.

Formula: AD/DB = AE/EC ⇒ DE ∥ BC

ABCDE3624DE ∥ BC ?

In the figure: 3/6 = 2/4 = 1/2, so DE ∥ BC.

Rule 1: Compare the ratios — Simplify both ratios and see if they match.

  • Easy: AD = 3, DB = 6, AE = 2, EC = 4: 1/2 = 1/2, parallel.
  • Medium: AD = 2, DB = 3, AE = 3, EC = 5: 2/3 ≠ 3/5, not parallel.
  • Exam-type: AD = 6, DB = 9, AE = 4, EC = 6: 2/3 = 2/3, parallel.

Rule 2: Using the whole side — AD/AB = AE/AC gives the same conclusion.

  • Easy: AD = 2, AB = 6, AE = 3, AC = 9: 1/3 = 1/3, parallel.
  • Medium: AD = 3, AB = 8, AE = 2, AC = 6: 3/8 ≠ 1/3, not parallel.
  • Exam-type: Unequal ratios always mean the line is not parallel.
Remember

The converse needs the ratios to be equal on both sides.

Common mistake

Concluding "parallel" after checking only one ratio.

Memory clue

"Equal ratios on both sides: line is parallel."

Practice questions

Easy
  1. 1In △UQK, F lies on UQ and X lies on UK. UF = 20 cm, FQ = 15 cm, UX = 4 cm and XK = 5 cm. Is FX ∥ QK?
  2. 2In △QHN, W lies on QH and C lies on QN. QW = 12 cm, WH = 2 cm, QC = 36 cm and CN = 9 cm. Is WC ∥ HN?
  3. 3In △YNC, D lies on YN and L lies on YC. YD = 6 cm, DN = 36 cm, YL = 3 cm and LC = 18 cm. Is DL ∥ NC?
Medium
  1. 4In △AHD, W lies on AH and G lies on AD. AW = 6 cm, WH = 2 cm, AG = 6 cm and GD = 4 cm. Is WG ∥ HD?
  2. 5In △VUP, D is on VU and T is on VP with VU = 32 cm, VD = 20 cm, VP = 32 cm and VT = 20 cm. Is DT ∥ UP?
  3. 6In △HEX, Z is on HE and G is on HX with HE = 28 cm, HZ = 8 cm, HX = 35 cm and HG = 10 cm. Is ZG ∥ EX?
Challenging
  1. 7In △BVZ, R is on BV and H is on BZ with BV = 30 cm, BR = 25 cm, BZ = 19 cm and BH = 16 cm. Is RH ∥ VZ?
  2. 8In △VAF, M is on VA and B is on VF with VA = 10 cm, VM = 2 cm, VF = 30 cm and VB = 6 cm. Is MB ∥ AF?
Show answers (try them yourself first)
  1. 1No
  2. 2No
  3. 3Yes
  4. 4No
  5. 5Yes
  6. 6Yes
  7. 7No
  8. 8Yes

Equivalent forms of the ratios

From daily life

If a 10 m pole is cut into 4 m and 6 m, then the part-to-whole ratios 4/10 and 6/10 tell the same story as 4/6.

Simple definition

From AD/DB = AE/EC you can also write AD/AB = AE/AC, DB/AB = EC/AC, and flipped forms such as AB/AD = AC/AE. All say the same thing.

Formula: AD/AB = AE/AC

AD/DB = AE/EC+ 1 both sidesAB/DB = AC/EC

In the figure: Add 1 to both sides: (AD + DB)/DB = (AE + EC)/EC, that is AB/DB = AC/EC.

Rule 1: Part to whole — Use AB = AD + DB and AC = AE + EC.

  • Easy: AD = 2, DB = 4, AE = 3: AD/AB = 2/6, so AC = 9, EC = 6.
  • Medium: AD = 3, DB = 3, AE = 4: AD/AB = 1/2, so AC = 8.
  • Exam-type: DB = 4, AB = 10, EC = 6: DB/AB = EC/AC gives AC = 15.

Rule 2: Flipped forms — Flipping both ratios keeps the equation true.

  • Easy: AB = 12, AD = 4, AC = 15: AB/AD = AC/AE gives AE = 5.
  • Medium: AB = 15, DB = 5, AC = 12: AB/DB = AC/EC gives EC = 4.
  • Exam-type: All forms are the same equation written differently.
Remember

Use whichever form has the two known lengths on one side.

Common mistake

Mixing a part-to-part ratio with a part-to-whole ratio in one equation.

Memory clue

"Same kind on the left and on the right."

Practice questions

Easy
  1. 1In △UGQ, SR ∥ GQ and US : SG = 8 : 6. Find RQ : UQ.
  2. 2In △CHD, SV ∥ HD and CS : SH = 1 : 2. Find CH : CS.
  3. 3In △ZBS, MF ∥ BS and ZM : MB = 4 : 8. Find ZF : ZS.
Medium
  1. 4In △ZHD, TV ∥ HD and ZT : TH = 2 : 7. Find ZH : TH.
  2. 5In △TWY, NL ∥ WY and TN : NW = 4 : 1. Find LY : TY.
  3. 6In △YGD, UQ ∥ GD and YU : UG = 8 : 2. Find YQ : YD.
Challenging
  1. 7In △LKN, VT ∥ KN and LV : VK = 4 : 8. Find LK : VK.
  2. 8In △FNQ, AB ∥ NQ and FA : AN = 5 : 2. Find BQ : FQ.
Show answers (try them yourself first)
  1. 13 : 7
  2. 23 : 1
  3. 31 : 3
  4. 49 : 7
  5. 51 : 5
  6. 64 : 5
  7. 73 : 2
  8. 82 : 7

Algebraic problems with the theorem

From daily life

When lengths are given as expressions in x, the theorem turns the picture into an equation to solve.

Simple definition

Put the lengths in AD/DB = AE/EC, cross-multiply, simplify (the x² often cancels), solve, and check that every length is positive.

ratioequationcrossmultiplysolve xchecklengths > 0

In the figure: x/(x − 2) = (x + 2)/(x − 1) gives x² − x = x² − 4, so x = 4.

Rule 1: Form and solve — Expand both sides and collect x terms.

  • Easy: AD = x, DB = x − 2, AE = x + 2, EC = x − 1: x = 4.
  • Medium: AD = x, DB = 6, AE = x + 2, EC = 9: 9x = 6x + 12, x = 4.
  • Exam-type: AD = 2x − 1, DB = 3, AE = x, EC = 2: 4x − 2 = 3x, x = 2.

Rule 2: Check the answer — Put x back and see that both ratios match and no length is negative.

  • Easy: x = 4: lengths 4, 2, 6, 3 give 4/2 = 6/3 = 2.
  • Medium: x = 2: lengths 3, 3, 2, 2 give 3/3 = 2/2 = 1.
  • Exam-type: A root making a length zero or negative is rejected.
Remember

Lengths are positive, so an answer giving a negative length is rejected.

Common mistake

Expanding (x − 2)(x + 2) wrongly; it equals x² − 4.

Memory clue

"Cross-multiply, cancel x squared, solve, check."

Practice questions

Easy
  1. 1In △WXQ, EM ∥ XQ, WE = 6 cm, WM = 3 cm and MQ = 9 cm. Find EX (in cm).
  2. 2In △BMP, NH ∥ MP, BN = 4 cm, BH = 8 cm and HP = 6 cm. Find NM (in cm).
  3. 3In △ZGD, XA ∥ GD, ZX = 20 cm, ZA = 20 cm and AD = 30 cm. Find XG (in cm).
Medium
  1. 4In △ZVE, YS ∥ VE with ZY = x + 8, YV = x − 1, ZS = x and SE = x − 3 (all in cm). Find x.
  2. 5In △TSK, RP ∥ SK with TR = x + 12, RS = x + 8, TP = x + 4 and PK = x + 2 (all in cm). Find x.
  3. 6In △RWY, US ∥ WY with RU = x + 1, UW = x + 10, RS = x and SY = x + 8 (all in cm). Find x.
Challenging
  1. 7In △PVC, TU ∥ VC with PT = x + 5, TV = x + 7, PU = x and UC = x + 1 (all in cm). Find x.
  2. 8In △FND, KC ∥ ND with FK = x, KN = x − 2, FC = x + 11 and CD = x + 7 (all in cm). Find x.
Show answers (try them yourself first)
  1. 118
  2. 23
  3. 330
  4. 44
  5. 54
  6. 68
  7. 75
  8. 811

Mid-points, trapeziums and other uses

From daily life

A shelf at the half-way height of a triangular roof is exactly parallel to the base, and the same idea works in a trapezium.

Simple definition

If D and E are mid-points of AB and AC, then AD/DB = AE/EC = 1, so DE ∥ BC. In a trapezium ABCD with AB ∥ DC, a line parallel to AB cuts the legs in the same ratio, and the diagonals satisfy AO/OC = BO/OD.

Formula: AO/OC = BO/OD

ABCDEmid-points: DE ∥ BC

In the figure: Equal tick marks show D and E are mid-points, so DE ∥ BC.

Rule 1: Mid-points — Equal parts on two sides mean the joining line is parallel to the third.

  • Easy: AB = 12, D mid-point: AD = 6, DB = 6.
  • Medium: AD = DB and AE = EC: DE ∥ BC.
  • Exam-type: BC = 14: the mid-point line DE = 7.

Rule 2: Trapezium and diagonals — Treat the parallel line as the line in the triangle theorem.

  • Easy: AB ∥ DC, EF ∥ AB: AE = 3, ED = 6, BF = 4 gives FC = 8.
  • Medium: Diagonals meet at O: AO = 6, OC = 3, BO = 4 gives OD = 2.
  • Exam-type: AO = 10, OC = 5, OD = 3 gives BO = 6.
Remember

In every trapezium problem find the two parallel sides first.

Common mistake

Using the theorem when the line is not parallel to any side.

Memory clue

"Parallel sides first, then ratios."

Practice questions

Easy
  1. 1In △LMV, E is the mid-point of LM and EU ∥ MV, with U on LV. If LV = 24 cm, find LU (in cm).
  2. 2In △KGS, F is the mid-point of KG and FU ∥ GS, with U on KS. If KS = 52 cm, find KU (in cm).
  3. 3In △BHK, L is the mid-point of BH and LN ∥ HK, with N on BK. If BK = 12 cm, find BN (in cm).
Medium
  1. 4In △EUZ, HW ∥ UZ, EH = 11 cm and HU = 2 cm. In what ratio does W divide EZ (that is, EW : WZ)?
  2. 5In △HDV, FG ∥ DV, HF = 6 cm and FD = 12 cm. In what ratio does G divide HV (that is, HG : GV)?
  3. 6In trapezium MZAU, MZ ∥ UA and the diagonals meet at F. If FM = 12 cm, FA = 10 cm, FU = 25 cm, find FZ (in cm).
Challenging
  1. 7In trapezium NFUQ, NF ∥ QU and the diagonals meet at G. If GU = 24 cm, GF = 18 cm, GQ = 36 cm, find GN (in cm).
  2. 8In trapezium SPZU, SP ∥ UZ and the diagonals meet at E. If ES = 16 cm, EP = 16 cm, EU = 8 cm, find EZ (in cm).
Show answers (try them yourself first)
  1. 112
  2. 226
  3. 36
  4. 411 : 2
  5. 51 : 2
  6. 630
  7. 712
  8. 88