Question

Given a triangle with unequal sides, if P is the set of all points which are equidistant from B and C, and Q is the set of all points which are equidistant from sides AB and AC, then what is the P intersection with Q equal to?

Answer 1

Hint- In order to deal with this question first we will make the clean diagram of the given problem further we will use the definition as the incenter I of triangle ABC is equidistant from all 3 sides AB, AC, & BC and we will find the intersection point according to it.

Complete step-by-step answer:
We will first make the correct diagram of the given problem, as shown below

Here H is the intersection point of P and Q.

Since ABC scalene triangle.

And P is a group of points that are equidistant from B & C

The perpendicular bisector of BC lies in any point belonging to Set P. Can extend this perpendicular bisector to infinity.

 Cardinal (Set P) = Infinite elements

Now, P is equidistant from all three sides of AB, AC, & BC

Since set Q, contains elements, which are equidistant from side AB & AC.

Thus locus Q is the angle bisector of < A.

So, every point of angle bisector of ∠A , will be equidistant from AB, & AC.

So, cardinal (Set Q) = endless elements

Now we'll consider both sets in common

Therefore, the common element(point) is the intersection point of the perpendicular bisector of BC & angle bisector of BA

Therefore, PnQ = the point of intersection of BC's perpendicular bisector & angle bisector of affordable A. And since triangle is triangle with scalene,

Consequently, Cardinal (PnQ) =1

Note- The triangle inequality states that the sum of the lengths of a triangle on any two sides must be greater than or equal to the length of the third side.