Question

If the centroid and circumcentre of a Triangle are (3,3) and (6,2) respectively Then, the orthocentre is​

Answer 1

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.

Answer 2

${\pmb{\sf{\underline{Given \; that...}}}}$

★ The centroid and circumcentre of a triangle are (3,3) and (6,2) respectively.

${\pmb{\sf{\underline{To \; find...}}}}$

★ The orthocentre if the centroid and circumcentre of a triangle are (3,3) and (6,2) respectively.

${\pmb{\sf{\underline{Solution...}}}}$

★ The orthocentre = (-3,5)

${\pmb{\sf{\underline{Using \; Concept...}}}}$

$\underline{\bigstar\:\textsf{Section Formula\; :}}$

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.

${\underline{\boxed{\sf{\quad \Big(x, y \Big) = \Bigg(\dfrac{mx_2 + nx_1}{m + n} \dfrac{my_2 + ny_1}{m + n}\Bigg) \quad}}}}$ ⠀

${\pmb{\sf{\underline{Knowledge \; Required...}}}}$

★ The centroid divides the median in the ratio of 2:1

${\pmb{\sf{\underline{Assumptions...}}}}$

★ Let us assume the orthocentre of that triangle as x and y. (according to the formula)!

${\pmb{\sf{\underline{Full \; Solution...}}}}$

~ 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.

${\underline{\boxed{\sf{\quad \Big(x, y \Big) = \Bigg(\dfrac{mx_2 + nx_1}{m + n} \dfrac{my_2 + ny_1}{m + n}\Bigg) \quad}}}}$ ⠀

$\sf :\implies \Big(x, y \Big) = \Bigg(\dfrac{mx_2 + nx_1}{m + n} \dfrac{my_2 + ny_1}{m + n}\Bigg) \\ \\ \sf :\implies \Big(x, y \Big) = \Bigg(\dfrac{(2)(6) + (1)(x)}{2 + 1} \dfrac{(2)(2) + (1)(y)}{2 + 1}\Bigg) \\ \\ \sf :\implies \Big(x, y \Big) = \Bigg(\dfrac{12 + x}{3} \dfrac{4 + y}{3}\Bigg)$

~ Now by using the above dimension and as the value of the centroid is given we just have to put it as mentioned below!

$\sf :\implies \Big(x, y \Big) = \Bigg(\dfrac{12 + x}{3} \dfrac{4 + y}{3}\Bigg) \\ \\ \sf :\implies \Big(3,3 \Big) = \Bigg(\dfrac{12 + x}{3} \dfrac{4 + y}{3}\Bigg) \\ \\ \sf :\implies \Big(3,3 \Big) = \Bigg(\dfrac{12x}{3} \dfrac{4y}{3}\Bigg) \\ \\ \sf :\implies \Big(9 \Big) = \Bigg(12 + x \: \: \: \: 4 + y\Bigg) \\ \\ \sf :\implies\Bigg(x = 9 - 12 \: \: \: \: y = 9 - 4\Bigg) \\ \\ \sf :\implies\Bigg(x = - 3 \: \: \: \: y = 5\Bigg)$

Henceforth, (-3,5) is the orthocentre

${\pmb{\sf{\underline{Additional \; Knowledge...}}}}$

$\underline{\bigstar\:\textsf{Distance Formula\; :}}$

Distance formula is used to find the distance between two given points.

${\underline{\boxed{\sf{\quad Distance = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \quad}}}}$ ⠀

$\underline{\bigstar\:\textsf{Section Formula\; :}}$

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.

${\underline{\boxed{\frak{\quad \Big(x, y \Big) = \Bigg(\dfrac{mx_2 + nx_1}{m + n} \dfrac{my_2 + ny_1}{m + n}\Bigg) \quad}}}}$ ⠀

$\underline{\bigstar\:\textsf{Mid Point Formula\; :}}$

Mid Point formula is used to find the Mid points on any line.

${\underline{\boxed{\frak{\quad \Bigg(\dfrac{x_1 + x_2}{2} \; or\; \dfrac{y_1 + y_2}{2} \Bigg)\quad}}}}$