Question

Using ruler and compasses only,

1. Construct a triangle ABC with the following data:
   Base AB = 6 cm, BC = 7.5 cm and angle CAB = 60°. 
2. In the same diagram, draw a circle which passes through the points A, B and C and mark its centre O. 
3. Draw a perpendicular from O to AB which meets AB in D.
4. Prove that: AD = BD.

Answer 1

Steps of construction:

1. Draw a line segment AB = 6 cm.
2. At A, draw a ray (AX) making an angle of 60° with AB.
3. With B as center and radius = 7.5 cm draw an arc which intersects AX ray at C.
4. Join BC. ABC is the required triangle.
5. Draw the perpendicular bisectors of AB and AC intersecting each other at O.
6. With center O and radius OA, OB or OC, draw a circle passing through A, B and C.
7. From O, draw OD ⊥ AB.

Proof:

In right △OAD and △OBD,

⇒ OA = OB (Radius of same circle)

⇒ OD = OD (Common)

⇒ ∠ODA = ∠ODB (Both = 90°)

∴ △OAD ≅ △OBD (By R.H.S. axiom)

By C.P.C.T.

⇒ AD = BD.

Hence, proved that AD = BD.