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comment upon the nature of triangleQ.14 A triangle has side lengths 18,24 and 30. Find the area of the triangle whose vertices are the incentre,circumcentre and centroid of the triangle.fa rectanale The other two vertices lie on the line |
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Answer» 18,24 and 30 are the sides of a right triangle, because 18^2 + 24^2 = 30^2 consider in a x-y plane pointsO(0,0), A(18,0), B(0,24) circumcenter of a right triangle is the midpoint F of hypotenuse AB(coordinates of the midpoint of a segment are the mean of the coordinates of its vertices)F(9,12) centroid G of any triangle has coordinates which are the mean of the coordinates of triangle's vertices, G(6,8) incenter H is the center of inscribed circle, whose radius isr = area/half-perimeter area = (18x24)/2 = 216half-perimeter = (18 + 24 + 30)/2 = 36 r = 216/36 = 6 as inscribed circle is tangent to both axis, incenter H has coordinates both equal to circle's radiusH(6,6) Requested triangle has verticesF(9,12), G(6,8), H(6,6) |
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