1. Chapter 11 Class 10 Constructions
2. Concept wise

Transcript

Draw a pair of tangents to a circle of radius 5 cm which are inclined to each other at an angle of 60°. Concept Given angle between tangents is 60° i.e. ∠ QPR = 60° ⇒ ∠ OQR = 2 × 60° = 120° So, we need to draw ∠ OQR = 120° Also, OQ ⊥ QP & OR ⊥ PR Thus, to make tangents, we draw perpendiculars from point Q and R   Ex 11.2, 4 Draw a pair of tangents to a circle of radius 5 cm which are inclined to each other at an angle of 60°. The tangents can be constructed in the following manner: Draw a circle of radius 5 cm with centre O. We need make angle of 120° at center Draw diameter QOA Draw ∠ OAR = 60° Thus, ∠ QOR = 180° – ∠ OAR = 180 – 60 = 120° Therefore, our two tangents will touch the circle at Q and R Now, we know that tangent is perpendicular to radius. Draw perpendicular from point Q and point R Let P be point where both perpendiculars  intersect Thus, PQ and PR are the required tangents at angle of 60° Justification We need to prove PQ and PR are tangents and ∠ QPR = 60° Since PQ is perpendicular to OQ (radius) PQ must be a tangent Also, PR is a tangent Now, we need to prove ∠ QPR = 60° Since PQ ⊥ OQ, ∠ OQP = 90° Also, PR ⊥ OR,  ∠ ORP = 90° and ∠ QOR = 120° Now, in quadrilateral OQPR ∠ QPR + ∠ PQO + ∠ PRO + ∠ QOR = 360° ∠ QPR + 90° + 90° + 120°  = 360° ∠ QPR + 300°  = 360° ∠ QPR = 360° – 300° ∠ QPR = 60° Thus, angle between tangents is 60° Now, in quadrilateral OQPR ∠ QPR + ∠ PQO + ∠ PRO + ∠ QOR = 360° ∠ QPR + 90° + 90° + 120°  = 360° ∠ QPR + 300°  = 360° ∠ QPR = 360° – 300° ∠ QPR = 60° Thus, angle between tangents is 60°