The triangle ABC is isosceles and AB = BC = 5. D is the midpoint of AC and BD = 4.

Find the lengths of the tangents from A, B and D to the circle inscribed in the triangle ABD.

AD $= 3$      (A1)

Let the lengths of the tangents be as shown.

Then,

$x + y = 3$

$y + z = 4$

$x + z = 5$     (M1)A1

Solving,

$x = 2$ , $y = 1$ , $z = 3$     A1A1A1

[6 marks]