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+2 votes

Ans: 2 cm

From the center of the circle, draw a radius (len = 2.5cm) to point D. Now you get a right angled triangle with the sides : x cm, 1.5 cm and 2.5 cm. Using Pythogoras, you get x^2 = 2.5^2 - 1.5^2 which gives x = 2 cm.

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+1 vote

i have seen the other two answers. therefore another simple solution is as under:
if we complete the circle and draw mirror of x, applying one theorem of geometry x*x=4*1 or x=2

No relationship with geometric mean here. And why not (4x1)/2 ?

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0 votes

Let us consider that the centre of the circle lies between A and Let it be O
Then join O and D
In the triangle DOB
OB =(2.5-1)
Do=radius=2.5cm
Therefore applying Pythagoras theorem
X^2+1.5^2=2.5^2
Solving we get x= 2cm

In triangleABC, side AB = 20, AC = 11, and BC = 13. Find the diameter of the semicircle inscribed in ABC, whose diameter lies on AB, and that is tangent to AC and BC.

Triangle ABC has sides AB = 8, BC = 7, and AC = 9.
The segments AB and AC are extended and a circle O is constructed exterior to the triangle and is tangent to BC at D and the extended lines through AC and AB at N and M, as shown.
What is the radius of circle O?