If the centroid and circumcentre of a Triangle are (3,3) and (6,2) respectively Then, the orthocentre is
If the centroid and circumcentre of a Triangle are (3,3) and (6,2) respectively Then, the orthocentre is
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EXPLANATION.
Centroid of a triangle = (3,3).
Circumcentre of a triangle = (6,2).
As we know that,
Let we assume that,
orthocentre of a triangle = (x, y).
Note : Centroid (G) divides median in the ratio of 2 : 1.
As we know that,
Section formula :
⇒ x = mx₂ + nx₁/m + n
⇒ y = my₂ + ny₁/m + n.
Using this formula in equation, we get.
⇒ x = [(2)(6) + (1)(x)]/2 + 1.
⇒ x = [12 + x]/3
⇒ y = [(2)(2) + (1)(y)]/2 + 1.
⇒ y = [4 + y]/3.
Equate this equation with centroid, we get.
⇒ (12 + x)/3 , (4 + y)/3 = (3,3).
⇒ 12 + x/3 = 3
⇒ 12 + x = 9.
⇒ x = 9 - 12.
⇒ x = -3.
⇒ 4 + y/3 = 3.
⇒ 4 + y = 9.
⇒ y = 9 - 4.
⇒ y = 5.
Orthocentre of a triangle = (-3,5).
MORE INFORMATION.
Important notes :
(1) = If a triangle is right angled, then its Circumcentre is the mid-point of hypotenuse.
(2) = If a triangle is right angled triangle, then ortho Centre is the point where right angle is formed.
(3) = If the triangle is equilateral, then centroid, incentre, orthocentre, Circumcentre coincides.
(4) = orthocentre, centroid and Circumcentre are always collinear and centroid divides the line joining orthocentre and Circumcentre in the ratio = 2 : 1.
(5) = In an isosceles triangle centroid, orthocentre, incentre, Circumcentre lies on the same line.
★ The centroid and circumcentre of a triangle are (3,3) and (6,2) respectively.
★ The orthocentre if the centroid and circumcentre of a triangle are (3,3) and (6,2) respectively.
★ The orthocentre = (-3,5)
Section Formula is used to find the coordinates of the point(Q) which divides the line segment joining the points (B) and (C) internally or externally.
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★ The centroid divides the median in the ratio of 2:1
★ Let us assume the orthocentre of that triangle as x and y. (according to the formula)!
~ Question is easy, we just have to use the formula here. Firstly let us put the given values according to the formula as in the Equation form.
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~ Now by using the above dimension and as the value of the centroid is given we just have to put it as mentioned below!
Henceforth, (-3,5) is the orthocentre
Distance formula is used to find the distance between two given points.
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Section Formula is used to find the coordinates of the point(Q) which divides the line segment joining the points (B) and (C) internally or externally.
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Mid Point formula is used to find the Mid points on any line.