Let ABC be a right triangle in which AB = 6 cm, BC = 8 cm and ∠B = 90°. BD is the perpendicular from B on AC. The circle through B, C, and D is drawn. Construct the tangents from A to this circle. Give the justification of the construction.
Consider the following situation. If a circle is drawn through B, D, and C, BC will be its diameter as ∠BDC is of measure 90°. The centre E of this circle will be the mid-point of BC.
The required tangents can be constructed on the given circle as follows.
Join AE and bisect it. Let F be the mid-point of AE.
Taking F as centre and FE as its radius, draw a circle which will intersect the circle at point B and G. Join AG.
AB and AG are the required tangents.
The construction can be justified by proving that AG and AB are the tangents to the circle. For this, join EG.
∠AGE is an angle in the semi-circle. We know that an angle in a semi-circle is a right angle.
∴ ∠AGE = 90°
⇒ EG ⊥ AG
Since EG is the radius of the circle, AG has to be a tangent of the circle.
Already, ∠B = 90°
⇒ AB ⊥ BE
Since BE is the radius of the circle, AB has to be a tangent of the circle.