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Angle bisectors of triangle ABC with right angle at B meet at a point

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Angle bisectors of triangle ABC with right angle at B meet at a point  [#permalink]

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New post 26 Jul 2017, 14:30
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Internal angle bisectors of triangle ABC with right angle at B meet at a point P inside the triangle.What is the perpendicular distance from point P to side AB?
1) Length of side AB and BC is 8 cm and 6 cm respectively
2) The perimeter of the triangle ABC is 24 cm

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Re: Angle bisectors of triangle ABC with right angle at B meet at a point  [#permalink]

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New post 29 Jul 2017, 08:29
Angle bisectors meet at a point called Incenter
Hence the the perpendicular line drawn from Incenter to any side is the "Inradius" r. A circle with radius r can be inscribed in the triangle and we need to find the Inradius of the triangle

Approach 1: -
Statement 1: -
From AB and BC , we can find out the hypotenuse AC that is 10 units(By pythogoras theorem)
Area of Triangle is also given by the formula
r*s where s is the semiperimeter (a+b+c/2)......................(1)
s=6+8+10/2=12

Now area can also be found out by
1/2*base*height.............................................................(2)

Equating 1 and 2

1/2*6*8=r*12
r=2
Hence statement 1 is sufficent

Statement 2: -
Perimeter is given but no information for any of the sides is given, hence this is insufficient

Answer option A

Approach 2: -
Statement 1: -
In a right angled triange, Inradius,r, is also given by the formula a+b-c/2 (where a and b are the legs and c is the hypotenuse)
r=6+8-10/2=2

Hence r=2 & statement 1 is sufficent

Statement 2: - Perimeter is given but no information for any of the sides is given, hence this is insufficient
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Re: Angle bisectors of triangle ABC with right angle at B meet at a point   [#permalink] 29 Jul 2017, 08:29
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