Construct a circle of radius 5 cm draw two tangents to the circle perpendicular to each other

Last updated at Aug. 5, 2021 by

Ex 11.2, 4 (Concept) Draw a pair of tangents to a circle of radius 5 cm which are inclined to each other at an angle of 60°. Given angle between tangents is 60° i.e. ∠ QPR = 60° Since Angle at center is double the angle between tangents ∴ ∠ OQR = 2 × 60° = 120° So, we need to draw ∠ QOR = 120° ∴ We draw a radius, then second radius at 120° from first. Also, Tangent is perpendicular to radius So, OQ ⊥ QP & OR ⊥ PR Thus, to make tangents, we draw perpendicular from point Q and R So, we draw 90° from point Q and point R Thus, our figure will look like 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°. Steps of construction Draw a circle of radius 5 cm Draw horizontal radius OQ 3. Draw angle 120° from point O Let the ray of angle intersect the circle at point R Now, draw 90° from point Q 5. Draw 90° from point R 6. Where the two arcs intersect, mark it as point P ∴ PQ and PR are the tangents at an angle of 60° Justification We need to prove that PQ and PR are the tangents to the circle at angle of 60° . Since ∠ PQO = 90° ∴ PQ ⊥ QO Since tangent is perpendicular to radius, and QO is the radius ∴ PQ is the tangent to the circle Similarly, PR is the tangent to the circle Now, we prove ∠ P = 60° In quadrilateral PQOR Sum of angles = 360° ∠ P + ∠ Q + ∠ R + ∠ QOR = 360° ∠ P + 90° + 90° + 120° = 360° ∠ P + 180° + 120° = 360° ∠ P + 300° = 360° ∠ P = 360° – 300° ∠ P = 60° So, PQ and PR are tangents at an angle of 60°

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Last updated at July 14, 2020 by Teachoo

Ex 11.2, 5 Draw a line segment AB of length 8 cm. Taking A as centre, draw a circle of radius 4 cm and taking B as centre, draw another circle of radius 3 cm. Construct tangents to each circle from the centre of the other circle. Steps of construction Draw line segment AB of length 8 cm Taking A as center, draw a circle of radius 4 cm Taking B as center, draw a circle of radius 3 cm Now, we need to draw tangent from point A to the right circle, and from point B to the left circle. To draw tangents to the right circle, we need to draw perpendicular bisector of line AB 4. Make perpendicular bisector of AB Let M be the midpoint of AB. 5. Taking M as center and MA as radius, draw a circle. 6. Let blue circle intersect left circle at P, Q Let blue circle intersect right circle at R, S Join BP, BQ, AR and AS ∴ AR, AS and BP, BQ are the required tangents Justification We need to prove that BP, BQ, AR, AS are the tangents to the circle. Join OP, OQ, OR and OS Now, ∠APB is an angle in the semi-circle of the blue circle And we know that, Angle in a semi-circle is a right angle. ∴ ∠ABP = 90° ⇒ AP ⊥ BP Since AP is the radius of the circle, BP has to be a tangent of the circle. Similarly, we can prove BQ, AR, AS are tangents of the circle.