Question

In a right angled triangle ABC right angled at B, A circle is inscribed in it and touches the triangle at points P Q & R, if the hypotenuse of triangle I.e..AC is 13cm, then find radius of circle?​

Answer 1

So we are given that the circle with the Radius 4cm that touch all the sides of the triangle.
Hence, it is an incircle.
And also the radius will be perpendicular to the sides, at the point of contact of the circle and the tangent side.
Let the point of contact of this circle with the sides AC,AB,BC be D,E,F.

And let the centre of the circle be O.
Now from the figure OEBF will be a square with side 4cms
Hence BO = 4√2cms.
and as given OD = 4cms.
therefore, BD = BO+OD = 4+4√2 cms.
also BD is the angular bisector of the angle ∟ABC.
Therefore , ∟ABD=45°.
and ∟BDA=90°('cuz the angle created by the radius and tangent at the point of contact of circle and its tangent is 90°).
Eventually, ∟DAB=45°.
So, ∆ABD will be isosceles right angled triangle.
The base, height and the hypotenuse of the triangle will be in the ratio 1:1:√2 respectively
Hence, in such(isosceles right angled triangle) cases hypotenuse will be
√2 x (base or height)
In this case , AB is the hypotenuse of the ∆ABD
So, AB = √2 x AD = √2 x BD
=>AB = √2 x (4+4√2)
=>AB = 8+4√2cms


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