Mathematics — Standard (041) — CBSE Class 10 board question

Steps of Construction:

1. Draw a line segment AB = 8 cm.
2. Since AP : PB = 1 : 3... wait — AP : PB = 1 : 5, so total parts = 1 + 5 = 6.
3. Draw a ray AX making an acute angle with AB.
4. Mark 6 equal points A₁, A₂, A₃, A₄, A₅, A₆ on AX such that AA₁ = A₁A₂ = … = A₅A₆.
5. Join A₆ to B.
6. Through A₁, draw a line parallel to A₆B meeting AB at point P.

Result: P is the required point on AB such that AP : PB = 1 : 5.

Verification: AP = $\dfrac{1}{6} \times 8 = \dfrac{4}{3}$ cm ≈ 1.33 cm; PB = $\dfrac{5}{6} \times 8 = \dfrac{20}{3}$ cm ≈ 6.67 cm. Thus AP : PB = 1 : 5. ✓

---

• The key idea: divide AB into (1+5) = 6 equal parts; P lies after the 1st part from A.
• The parallel line through A₁ to A₆B uses the Basic Proportionality Theorem (Thales Theorem) to ensure the correct ratio.
• Examiners award marks for: correct number of arcs (6), joining A₆B, drawing the parallel line through A₁, and labelling P clearly.
• Always write the verification step — it fetches the final mark.