Question

Draw a circle of radius 3 centimetres.
(a) Draw triangle ABC with this circle as circumcircle and angle ${50}^o,{60}^o$ and ${70}^o$.
(b) Construct triangle PQR, outside the circle, by drawing tangents to the circle at the points A, B and C.
(c) Find all angles of triangle PQR.

Answer 1

(a) Steps of construction:
1. Draw a circle with centre O and having radius 3 cm.
2. To make the three points A, B, and C in the circle, join A to the centre of the circle O.
3. If $m\angle BAC$ is to be ${50}^o,m\angle BOC$ should be ${100}^o$.
4. If $m\angle ABC$ is to be ${60}^o,m\angle AOC$ should be ${120}^o$.
5. If $m\angle ACB$ is to be ${70}^o,m\angle AOB$ should be ${140}^o$.
6. Join the points B and C such that $m\angle AOC={100}^o$ and $m\angle BOC={120}^o$>
Thus, $\Delta$ABC is the required triangle.
(b) Steps of construction:
1. Extend OA and draw perpendicular to it through A.
2. Extend OB and draw perpendicular to it through B.
3. In the same way draw perpendicular to OC through C.
Let the points of intersection of these perpendicular be P, Q and R, so we get the required $\Delta$PQR.
(c) In the quadrilateral PAOC,
$m\angle AOC={120}^o$
$\Rightarrow m\angle P={180}^o-{120}^o={60}^o$ .... (opposite angles of a quadrilateral are suplementary)
In the same way, $m\angle Q={180}^o-{140}^o={40}^o$
And, $m\angle R={180}^o-{100}^o={80}^o$