Chapter 11: Constructions

Key Constructions:
• Division of a line segment in given ratio
• Construction of similar triangles (scale factor)
• Construction of tangents to a circle from external point

Exercise 11.1

Q1. Divide a line segment of length 7.6cm in the ratio 5:8. Measure the two parts.

Solution

Steps of Construction:
1. Draw AB = 7.6cm
2. Draw ray AX making acute angle with AB
3. Mark 5+8=13 equal points on AX: A₁, A₂,...,A₁₃
4. Join A₁₃ to B
5. Draw line through A₅ parallel to A₁₃B meeting AB at C

AC = 5/13 × 7.6 ≈ 2.92 cm
CB = 8/13 × 7.6 ≈ 4.68 cm

Q2. Construct ΔABC with sides 4cm, 5cm, 6cm and then a similar triangle with ratio 2/3.

Solution

Steps:
1. Draw ΔABC: AB=4cm, BC=6cm, AC=5cm
2. Draw ray BX making acute angle with BC
3. Mark 3 equal points on BX (denominator=3)
4. Join 3rd point (B₃) to C
5. Draw line through 2nd point parallel to B₃C
6. This meets BC at C₂. Draw C₂A₂ || CA

New triangle sides: 8/3 cm, 10/3 cm, 4 cm

Q3. Construct ΔABC with sides 5cm, 6cm, 7cm and similar triangle with sides 7/5 of corresponding sides.

Solution

Steps:
1. Draw ΔABC: AB=5cm, BC=7cm, AC=6cm
2. Draw ray BX, mark 5 equal points
3. Join B₅ to C
4. Draw line through B₇ (extends beyond) parallel to B₅C
5. This meets BC extended at C₄
6. Draw C₄A₄ || CA to meet BA extended

New triangle sides: 7cm, 49/5 cm, 42/5 cm

Exercise 11.2

Q1. Draw circle radius 6cm. From point 10cm away, construct tangents. Measure lengths.

Solution

Steps:
1. Draw circle with centre O, radius 6cm
2. Mark point P such that OP=10cm
3. Bisect OP at M (midpoint)
4. Draw circle with centre M, radius MO
5. This circle meets given circle at A and B
6. Join PA and PB

PA=PB=√(10²-6²)=√64=8cm

Q2. Construct tangents to circle radius 4cm from point on concentric circle radius 6cm.

Solution

Steps:
1. Draw two concentric circles with centre O, radii 4cm and 6cm
2. Mark point P on outer circle
3. Bisect OP at M
4. Draw circle centre M radius MO
5. Meets inner circle at A and B
6. Join PA and PB

PA=PB=√(6²-4²)=√20=2√5 cm

Q3. Draw circle radius 3cm. Take point P 7cm from centre. Construct pair of tangents.

Solution

Steps:
1. Draw circle centre O, radius 3cm
2. OP = 7cm
3. Bisect OP at M
4. Draw circle centre M, radius 3.5cm
5. Meets given circle at A and B
6. PA and PB are required tangents

Length = √(7²-3²) = √40 = 2√10 cm

← Chapter 10: Circles Chapter 12: Areas Related to Circles →

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