Asked in: March 2024 board paperDraw a circle with centre O having radius 3 cm. Draw tangent segments PA and PB through the point P outside the circle such that \(\angle APB = 70^\circ\).

Answer: Draw radii OA and OB with \(\angle AOB = 110^\circ\); the tangents at A and B meet at P with \(\angle APB = 70^\circ\).

Analysis: In \(\square OAPB\), \(\angle OAP = \angle OBP = 90^\circ\) (tangent ⊥ radius). So \(\angle AOB = 360^\circ - 90^\circ - 90^\circ - 70^\circ = 110^\circ\).

Steps of construction:

1. Draw a circle with centre O and radius 3 cm.
2. Draw radius OA. At O draw ray OB making \(\angle AOB = 110^\circ\) with OA, B on the circle.
3. At A draw a line perpendicular to OA, and at B draw a line perpendicular to OB.
4. Let these lines meet at P. PA and PB are the required tangent segments, and \(\angle APB = 70^\circ\).

[Figure: circle with centre O, radius 3 cm; radii OA and OB with \(\angle AOB = 110^\circ\); tangents at A and B meeting at P, \(\angle APB = 70^\circ\).]

Check: PA = PB; each is \(3 \tan 55^\circ \approx 4.3\) cm.