Bunuel wrote:
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
Purely from the perspective of DSIf you have a UNIQUE triangle, you would be able to find almost all the related terms, be it angles, area, incenter, etc.Now, we are given a right angled triangle with B as 90.
(1) Length of side AB and BC is 8 cm and 6 cm respectively
AC is the hypotenuse and 10 in length, because sides are in ratio 6:8:10 or 3:4:5.
Fixed and unique triangle.
Sufficient
(2) The perimeter of the triangle ABC is 24 cm.
We can have various triangles as sides need not be integer.
Insufficient
A
PS solution.
When you join all internal angle bisectors as given in sketch, we have a point O, which is called incenter.
Incenter will give us the radius of the incircle.
Attachment:
Untitled02.png [ 31.21 KiB | Viewed 514 times ]
The perpendicular distance from point P to side AB is nothing but the radius r.(1) Length of side AB and BC is 8 cm and 6 cm respectively
So we have a 6:8:10 triangle.
Area of \(\triangle ABC = \frac{1}{2}*8*6=24\)...(i)
If we take individual inner triangles, Area of \(\triangle ABC = A(\triangle ABO)+A(\triangle ACO)+A(\triangle CBO)= \frac{1}{2}*8*r+\frac{1}{2}*r*6+\frac{1}{2}*10*r=\frac{1}{2}*24*r=12\)...(ii)
From i and ii, 12r=24 or r=2
Sufficient
(2) The perimeter of the triangle ABC is 24 cm
Various possibilities
Insufficient
A
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