Using ruler and compasses only construct a ΔABC having base = 5 cm, vertical angle = 45° and median through vertex equal to 4 cm. Draw the incircle of the triangle.

Steps of construction :

1. Draw a line segment BC = 5 cm.

2. Construct perpendicular bisector of BC.

3. At point B, construct an angle of 45°, ∠XBC = 45°.

4. Let this ray XB intersect the perpendicular bisector of BC at O. With a radius of OB draw major arc.

5. From M, draw an arc of radius 4 cm, that intersects the major arc and label it as A.

6. Join AB and AC. Hence, ABC is the required triangle.

7. Construct the angle bisector BH and CI of ∠ABC and ∠ACB respectively.

8. Let the two angle bisectors intersect at point P.

9. Draw PD perpendicular to BC. With P as centre and PD as radius, draw a circle.