The value of x in the given circle determines if the point P is the
circumcenter or the incenter of the triangle.
- The value of x that will make the P the incenter of the triangle is x = 7°
- The value of x, that makes P, the circumcenter of the triangle, is x = 6°
Reasons:
Incenter of a triangle;
The incenter of a triangle is the center of an inscribed circle of the triangle
Therefore, given that x makes P, the incenter of the triangle, we have;
(3·x + 3) = 24
Therefore;
[tex]x = \dfrac{24 - 3}{3} = 7[/tex]
Therefore;
The value of x that will make the P the incenter of the triangle is x = 7°
Circumcenter of the triangle;
The circumcenter of the triangle, is the center of the circle circumscribing
the triangle.
Therefore, the length of the segments from P, to the vertices are the length
of the radius of the circumscribing circle and are therefore equal.
Which gives;
5·x - 4 = 26
[tex]x = \dfrac{26 + 4}{5} = 6[/tex]
The value of x, that makes P, the circumcenter of the triangle, is x = 6°
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