The orthocentre for the triangle ABC is (1.85, 5.88).
Triangle ABC has vertices at A(2, 8), B(6, 2) and C(-3, 2).
We need to use analytical geometry to determine the coordinates of the orthocentre.
How to find the orthocentre using analytical geometry?
Steps to find the orthocentre:
Step 1: First, we will find the slopes of any two sides of the triangle (say AC and AB).
Step 2: Next, we can find the slopes of the corresponding altitudes. Remember that if two lines are perpendicular to each other, they satisfy the following equation.
Step 3: Next, we will use the slope-point form of the equation of a straight line to find the equations of the lines that are coincident with the altitudes BE and AD.
Step 4: Next, we can solve the equations of BE and CF simultaneously to find their solution, which gives us the coordinates of the orthocentre H.
Now, the slope of AC=2-8/-3-2=6/5 and the slope of BE=5/6.
The slope of AB=2-8/6-2=-6/4=-3/2 and the slope of CF=-2/3
The slope of BE=y-2/x-6=5/6
⇒6x-5y-26=0----(1)
The slope of CF=y-2/x+3=-2/3
⇒-2x-3y=0----(2)
By solving (1) and(2), we get
y=26/14=13/7=1.85 and x=5.88
Therefore, the orthocentre for the triangle ABC is (1.85, 5.88).
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