Draw a right triangle ABC in which AB = 6 cm, BC = 8 cm and ∠B = 90°. Draw BD perpendicular from B on AC and draw a circle passing through the points B, C and D. Construct tangents from A to this circle.

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.

#### Solution 1

Follow the given steps to construct the figure.**Step 1**

Draw a line BC of 8 cm length.**Step 2**

Draw BX perpendicular to BC.**Step 3**

Mark an arc at the distance of 6 cm on BX. Mark it as A.**Step 4**

Join A and C. Thus, ∆ABC is the required triangle.**Step 5**

With B as the centre, draw an arc on AC.**Step 6**

Draw the bisector of this arc and join it with B. Thus, BD is perpendicular to AC.**Step 7**

Now, draw the perpendicular bisector of BD and CD. Take the point of intersection as O.**Step 8**

With O as the centre and OB as the radius, draw a circle passing through points B, C and D.**Step 9**

Join A and O and bisect it. Let P be the midpoint of AO.**Step 10**

Taking P as the centre and PO as its radius, draw a circle which will intersect the circle at point B and G. Join A and G.

Here, AB and AG are the required tangents to the circle from A.

#### Solution 2

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.

**Step 1**

Join AE and bisect it. Let F be the mid-point of AE.

**Step 2**

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.

**Justification**

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.