Using ruler and compasses only, construct a triangle POR such that $ \angle P = 120^\circ $ , $ PO = 5\;cm $ and PR=6 cm. In the same figure find a point which is equidistant from its sides. Name this point. With this point as the centre, draw a circle touching all the sides of the triangle.
A.Circumcentre
B.Incentre
C.Mid point
D.Data insufficient

Hint: Using the knowledge of construction by ruler and scale to draw the required triangle and then find out the point which is equidistant from all the sides of the triangle by drawing angle bisectors of all the angles. Taking this point as the centre, draw a circle to find out what is the name of that point.

Complete step-by-step answer:
Construction of the triangle POR –
I.Draw a line PR of length 6 cm with the help of scale.
II.Make an angle of $ 120^\circ $ with the help of a compass.
III.Cut a line segment PO=5 cm.
IV.Join the points O and R, such that OR=9.6 cm
This way we have constructed the required triangle POR.
V.To find out the point equidistant from the three sides, we draw angle bisectors of all the angles of the triangle as shown in the figure.

V.Now, taking this point as the centre, we make a circle touching the three sides of the triangle.

The circle inside a triangle is called incircle and the centre of this circle is called incentre.
Thus point E is called the incentre.
So, the correct answer is “Option B”.

Note: Drawing shapes, angles or lines accurately is known as construction. These constructions use only compass, ruler and a pencil. No numbers are involved in construction that’s why it is called the true form of geometric construction. The basics of construction include drawing line segments, right angles and angle bisectors.