90+ marks in Unit 11 – easy constructions & answers!
HSA Notes Features
- 100% NBF/FBISE correct
- Ex 11.1 + 11.2 + Misc solved
- Step-by-step constructions
- Updated 2025 syllabus
- Circle centers, tangents, arcs
- Free PDF with diagrams
- Trusted for top scores
Target Audience
FBISE + Punjab boards. Top schools: Beaconhouse, Roots, LGS, City School, Karachi Grammar, Aitchison, Lawrence, ICG, APS, FPS.
Unit 11 Key Solved Highlights
Ex 11.1
Q1 Locate center: Perp bisectors of two chords intersect at O.
Q2 Center of arc XYZ: Perp bisectors of XY & YZ intersect at O.
Q3 Circle through A B C (AB=BC=3 cm, ∠B=120°): Perp bisectors of AB & BC intersect at O, r = OA.
Q4 Circle through P Q R (PQ⊥QR, PQ=4 cm, QR=3 cm): Perp bisectors intersect at O, r=2.5 cm.
Q5 Circle with round object: Measure diameter, r = diameter/2.
Q6 Center of arc DEF (DE=3.5 cm, EF=4.2 cm, ∠E=60°): Perp bisectors of DE & EF intersect at O.
Q7 Arc to full circle without center: Perp bisectors of two chords on arc intersect at O, complete circle.
Q8 Circular path d=50 m: (i) Perp bisectors of two chords → center (ii) r=25 m.
Q9 Arc tiles r=7 in, arc=2 in each: 22 tiles for full circle.
Q10 Boat swing: Path circle, center at fixed top point.
Ex 11.2
Q1 Divide arc: Perp bisector intersects arc at midpoint C.
Q2 Tangent at A without center: (i) Middle – chord through A, perp at A. (ii) End – chord AB, perp bisector to C, perp to AC at A.
Q3 Tangent from X to arc ABC: Perp bisectors find O, tangents from X to circle.
Q4 Circle r=4 cm, tangent at P: Radius OP, perp at P.
Q5 Circle r=2.5 cm, A 5.8 cm from O: Midpoint M of OA, circle r=MA intersects at P Q, AP AQ tangents.
Q6 Two tangents angle 45°/90°: Central angle 135°/90°, perps at P Q intersect at A.
Q7 Equal circles r=3 cm, centers 6 cm: (i) Direct – external division, tangents (ii) Transverse – internal division, tangents.
Q8 Circles r=2 cm 3.5 cm, centers 6.4 cm: Direct (external div 4:7), transverse (internal div 4:7).
Q9 Touching circles r=2.6 cm 3.6 cm: (i) Perp at contact (ii) Direct tangent (iii) Two common (direct & transverse).
Q10 Intersecting circles r=2.5 cm 3 cm, centers 4.8 cm: Two common tangents (direct & transverse).
Q11 Pulleys r=7 in 14 in, centers 28 in: Rope ≈120 in.
Q12 Satellite alt 7600 km, Earth r=6400 km: Distance ≈12450 km.
Q13 Two rods d=12 cm: Strap ≈61.7 cm.
Q14 Mirror r=20 cm, wire 30 cm: Hook 5 cm above top.
Misc Ex 11
i (a) 1
ii (c) perpendicular
iii (b) 2
iv (b) parallel
v (c) 8 cm
vi (a) 1
vii (d) congruent
viii (a) radii
ix (b) diameter
x (b) 1.1 cm
xi (d) central
xii (c) 90°
xiii (d) 50°
xiv (b) less than 180°
xv (c) greater than 180°
xvi (a) equal
xvii (c) 120°
xviii (c) equal to
xix (a) 45°
xx (d) 47°
Q2 ∠ABO=30°, ∠AOB=120°, ∠ACB=60°
Q3 ∠ABC=80°, ∠BCA=60°, ∠CAD=20°
Q4 ∠O=60°, ∠C=30°
View PDF for full diagrams & steps.
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