Question

Find the coordinates of the orthocentre of the triangle whose vertices are (1,2) ,(2,3) and (4,3).

Answer 1

Please  find below the answer.

Answer 2

The Coordinates of the Orthocentre are (1, 6)

Step-by-step explanation:

• Let A(1,2), B (2,3) & C(4,3) be the vertices of ABC

• Since the orthocentre of the triangle is the point of concurrence of the altitude from the vertices.

• Let, AD, BE & CF be the altitudes and O (h,k) be the orthocentre of the  triangle.

As AO⊥BC

$(Slope\ of\ AO) \times (Slope\ of\ BC)$ = -1  

⇒ $\frac {k-2}{h-1} \times (\frac {3-3}{4-2}) = -1$

⇒ $\frac {k-2}{h-1} \times (\frac {0}{2}) = -1$

⇒ h-1 = 0  

h = 1 """(1)

Also BO⊥AC

$(Slope\ of\ BO) \times (Slope\ of\ AC) = -1$

⇒ $\frac {k-3}{h-2} \times (\frac {3-2}{4-1}) = -1$

⇒ $\frac {k-3}{h-2} \times (\frac {1}{3}) = -1$

⇒ $k-3 = -3(h-2)$

⇒ $k-3 = -3h + 6$

⇒ $3h + k - 9 = 0$  """"(2)

Substituting h = 1 in (2) we get-

3(1) + k - 9 = 0

k-6 = 0  

k = 6  

So, The Coordinates of the orthocentre are (1, 6)