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Noteclerical

Constructions

Lesson 27 of 113

1. Ye topic hai kya?

Pencil se nahi — ruler + compass. Scale se length copy, compass se equal arcs. Paper: AB ko 3:5 mein C, given Δ se 2/5 similar, P se do tangents. Protractor se 90° banana allowed nahi jab theorem construction ho (tangent ⊥ radius compass se nikalta).

Yahan lock

m+n equal marks, parallel (BPT), scale <1 chhota / >1 bada, OP ka diameter-circle, angle in semicircle = 90° → tangent.

Yahan kya nahi

PA=PB length nikalna (Circles). Sector 60° area. Coordinate (m x₂+n x₁). Fold-punch. Incircle compass (IX extra) — yahan teen constructions.

2. Line segment — m : n internally

AB di, C chahiye taaki AC:CB = m:n. Section formula same ratio, lekin yahan draw. Equal compass-jumps + ek parallel (Thales).

Marks m+n (dono ka sum). Last mark ko B se jodo. m-th mark se us line ke parallel → AB pe C. AC:CB = m:n.
A B C A₃ A₈

3:5 → 8 equal arcs. A₈B join. A₃ se A₈B ke parallel (dashed) → C. AC:CB = 3:5.

Example

AB = 8 cm, C aisa ke AC:CB = 3:5. Kitne equal points ray pe?

Solution

  1. m+n

    3+5=8. Compass opening same, 8 jumps. Join 8th to B, 3rd se parallel.

  2. Nahi

    Sirf 3 marks (m) — last join ke liye n+m chahiye. Coordinate section (4,7) yahan nahi.

Answer8

3. Similar triangle — scale m/n

Given ΔABC. Naya triangle similar, corresponding sides m/n times. m<n → chhota (andar / side pe chhota). m>n → bada (extend).

Ray on a side (usually BC). Marks = max(m, n). Chhote ke liye: n marks, m-th se last-to-C ke parallel. Bade ke liye: m marks (m>n), n-th ko C se jod, m-th se parallel extend.
B C A C′ A′

Scale 2/5: ΔA′BC′ ~ ΔABC, BC′ = (2/5)BC. Chhota similar; bada original. AA similar — yahan draw.

Example

Scale 2/5. Ray BC pe kitne equal marks (chhota triangle)?

Solution

  1. n=5

    max(2,5)=5 marks. 5th ko C se jod, 2nd se parallel → C′. Phir C′ se CA ke parallel → A′.

Answer5

Scale 5/3 (bada): 5 marks, 3rd join C, 5th se parallel BC ke extend pe C′. Triangle original se bahar.

2/5 ko 5/2 mat ulta — sides 2.5 times nahi, 0.4 times. “Similar” Circles/Pythagoras nahi; BPT parallel se ratio.

4. External point se tangents

Circle centre O, radius r, P bahar. Do tangents PA, PB. Circles chapter mein length; yahan draw without protractor.

1 OP jodo. 2 OP ka midpoint M (perpendicular bisector). 3 Circle diameter OP (centre M, radius MO). 4 Yeh given circle ko A, B pe kaat-ti. 5 PA, PB tangents.
O M P A B

Dashed = diameter-OP circle. A, B intersection. PA, PB tangents. Bada = given circle. (r=7, PA=24 length Circles pe — yahan steps.)

Example

Is construction mein ∠OAP = 90° kyun?

Solution

  1. Semicircle

    A, diameter OP wale circle pe. Angle in semicircle: ∠OAP = 90°.

  2. Tangent

    OA radius, PA ⊥ OA → PA given circle ki tangent. (Class-IX semicircle + Class-X ⊥ theorem.)

AnswerAngle in semicircle, phir radius ⊥ tangent

5. Kyun — BPT aur semicircle

Divide / similar: equal marks ⇒ ray pe equal segments. Parallel ⇒ intercept theorem (BPT converse family) ⇒ AB pe same ratio. Similar triangle: do parallels (C′ aur A′) ⇒ AA similar.

Tangents: M midpoint ⇒ MO = MP = OP/2, A on both circles ⇒ OA = r, OA ⊥ PA. Do intersections ⇒ do tangents. P circle pe ho to diameter-circle trick ki zaroorat nahi: radius OP, uske ⊥ se ek tangent.

Compass opening change mat karo beech mein jab “equal marks” chal rahe. Parallel: corresponding arcs / copy-an-angle, protractor nahi.

6. Sawal — basic se pro

Q1 · Basic

AB ko 3:5 mein kaatna. Equal marks?

  1. 8
  2. 3
  3. 5
  4. 15

Solution

  1. m+n

    8. B sirf m. D product.

Answer8
Q2 · Basic

3:5 construction. Parallel kis mark se?

  1. 3rd, last-to-B ke parallel
  2. 5th
  3. 1st
  4. Midpoint of AB seedha

Solution

  1. m-th

    A₃ se A₈B ke parallel. D section formula, compass nahi.

AnswerA
Q3 · Basic

Scale 2/5 similar triangle. Result?

  1. Chhota, sides 2/5
  2. Bada, 5/2
  3. Congruent
  4. Area 2/5

Solution

  1. m<n

    Chhota. D: area (2/5)².

AnswerA
Q4 · Basic

Scale 2/5. Ray pe marks?

  1. 5
  2. 2
  3. 7
  4. 10

Solution

  1. max

    5. (Chhote case n=5.)

Answer5
Q5 · Medium

P bahar se tangents. Extra circle ka diameter?

  1. OP
  2. OA
  3. AB
  4. Given circle ka diameter

Solution

  1. Semicircle

    OP. Centre midpoint M.

AnswerOP
Q6 · Medium

Us extra circle se ∠OAP = 90° kaun si theorem?

  1. Angle in a semicircle
  2. BPT
  3. sin 30° = 1/2
  4. Equal tangents length (pehle se)

Solution

  1. IX

    Semicircle. D length Circles; construction ka reason 90° pehle.

AnswerA
Q7 · Medium

Scale 5/3. Triangle?

  1. Bada, sides 5/3
  2. Chhota 3/5
  3. Same size
  4. Impossible m>n

Solution

  1. m>n

    Extend. 5 marks, 3rd join C.

AnswerA
Q8 · Medium

P given circle pe. Tangents draw?

  1. Ek: radius OP, uske ⊥ at P
  2. Do, wahi diameter-OP trick
  3. Zero
  4. Protractor 45°

Solution

  1. On circle

    Exactly one. Perp to radius. Diameter circle P pe degenerate.

AnswerA
Q9 · Pro

Similar triangle scale 2/5. Ray BX pe equal marks kitne, parallel kis mark se?

  1. 5 marks; 2nd se, last-to-C ke parallel
  2. 2 marks; 2nd se
  3. 7 marks (2+5)
  4. Area 4:25 draw, construction nahi

Solution

  1. max(2,5)

    5 equal marks (badi denominator). 2nd mark se last (5th-to-C) ke parallel — intercept. Area k² Triangles pe, yahan steps.

AnswerA
Q10 · Pro

M, OP ka midpoint, kaise?

  1. Perpendicular bisector of OP (compass, do arcs)
  2. Scale se half guess
  3. Angle bisector of ∠AOB
  4. Join AB

Solution

  1. Compass

    O aur P se > OP/2 arcs, chord ke ⊥ bisector. B allowed nahi as “construction”.

AnswerA

Practice

Pehle khud, phir neeche kholo.

12345 678910

P1

2:7 mein divide. Marks?

  1. 9
  2. 2
  3. 7
  4. 14

Solution

  1. 2+7

    9. Parallel 2nd se.

Answer9
P2

Equal marks ka tool?

  1. Compass, same opening
  2. Protractor
  3. Set-square 45 only
  4. Freehand

Solution

  1. Arcs

    Same radius jumps.

AnswerA
P3

Scale 1 (m=n). Similar construction se?

  1. Congruent copy (same size)
  2. Half
  3. Impossible
  4. Circle

Solution

  1. k=1

    Same triangle (practical copy). Paper pe m≠n zyada.

AnswerA
P4

Diameter-OP circle given circle ko kitne useful points?

  1. 2 (A, B) jab P bahar
  2. 1 hamesha
  3. 0 hamesha
  4. 4

Solution

  1. Two tangents

    Do intersections. P andar ho to construction fail (0 tangents).

AnswerA
P5

Similar triangle mein parallel kis theorem se ratio deta?

  1. BPT / Thales
  2. Pythagoras
  3. tan θ = h/d
  4. PA = PB

Solution

  1. Parallel

    BPT. D Circles, construction ka reason nahi.

AnswerA
P6

5/3 similar. Marks on ray?

  1. 5
  2. 3
  3. 8
  4. 15

Solution

  1. max(5,3)

    5. Join 3rd to C, 5th se parallel.

Answer5
P7

OA ⊥ PA is construction ke baad kyun tangent?

  1. Radius ⊥ line at A ⇒ tangent
  2. Secant do points
  3. sin A = 21/29
  4. Fold paper

Solution

  1. Class-X

    Circles theorem apply after 90° mil jaye.

AnswerA
P8

3:5, C ke liye last join?

  1. A₈ to B
  2. A₃ to B
  3. A to B₈
  4. Mid to B

Solution

  1. Last mark

    A_{m+n}B. Parallel A₃ se.

AnswerA
P9

P andar circle ke. Diameter-OP method?

  1. Tangents nahi milenge (0)
  2. Do tangents
  3. Ek
  4. Infinite

Solution

  1. OP<r

    Circles: 0 tangents. Construction intersections useful nahi.

AnswerA
P10

2/5 similar, BC=10 cm. BC′?

  1. 4 cm
  2. 25 cm
  3. 5 cm
  4. 2 cm

Solution

  1. ×2/5

    4 cm. B ×5/2.

Answer4 cm

← Index · Circles

Agle topic: Areas related to circles.