Respuesta :
Answer:
(3, 0)
Step-by-step explanation:
The orthocentre of a triangle is the intersection point of the three perpendicular bisectors of the three sides.
A perpendicular bisector cuts a line in half and meets it at 90°.
You need to know the midpoint and slope for at least two of the sides.
Midpoint formula: [tex]M = (\frac{x_{2}+x_{1}}{2} ,\frac{y_{2}+y_{1}}{2})[/tex]
Slope formula: [tex]m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}[/tex]
When a line a perpendicular to another, its slope is the negative reciprocal.
The midpoint of the side is a point on the bisector.
Let's focus on the side AB.
Info set 1: A(0,0) x₁ = 0 y₁ = 0
Info set 2: B(3,3) x₂ = 3 y₂ = 3
Find the midpoint:
[tex]M = (\frac{x_{2}+x_{1}}{2} ,\frac{y_{2}+y_{1}}{2})[/tex]
[tex]M = (\frac{3+0}{2} ,\frac{3+0}{2})[/tex]
[tex]M = (\frac{3}{2} ,\frac{3}{2})[/tex]
[tex]M = (1.5, 1.5)[/tex]
Find the slope:
[tex]m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}[/tex]
[tex]m = \frac{3 - 0}{3 - 0}[/tex]
[tex]m = \frac{3}{3}[/tex]
[tex]m = 1/1[/tex]
To find the perpendicular line's slope, find the negative reciprocal. (Switch the top and bottom of the slope in fraction form, then change the negatuve/positive)
⊥m = -1/1
⊥m = -1
Find the equation of ⊥AB using m = -1 and (1.5, 1.5).
Substitute into the equation of a line.
y = mx + b
1.5 = -1(1.5) + b
1.5 = -1.5 + b Adding 1.5 to both sides
b = 3
⊥AB : y = -x + 3
Do the same for side AC
Info set 1: A(0,0) x₁ = 0 y₁ = 0
Info set 2: C(6,0) x₂ = 6 y₂ = 0
Find the midpoint:
[tex]M = (\frac{x_{2}+x_{1}}{2} ,\frac{y_{2}+y_{1}}{2})[/tex]
[tex]M = (\frac{6+0}{2} ,\frac{0+0}{2})[/tex]
[tex]M = (\frac{6}{2} ,\frac{0}{2})[/tex]
[tex]M = (3, 0)[/tex]
Find the slope:
[tex]m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}[/tex]
[tex]m = \frac{0 - 0}{6 - 0}[/tex]
[tex]m = \frac{0}{6}[/tex]
m = 0 This means the line is horizontal
A line perpendicular would be vertical. (⊥m = ∞)
Vertical lines are written in the form y=x, where x is the x-intercept.
Since the midpoint is (3, 0), the midpoint is 3.
⊥AC : y = 3
The intersection point of y = 3 and y = -x + 3 is the orthcentre.
Substitute y for 3 in the equation y = -x + 3
y = -x + 3
3 = -x + 3
x = 0
The orthocentre is (3, 0).
Check for the other line ⊥BC.
Info set 1: B(3,3) x₁ = 3 y₁ = 3
Info set 2: C(6,0) x₂ = 6 y₂ = 0
Find the midpoint:
[tex]M = (\frac{x_{2}+x_{1}}{2} ,\frac{y_{2}+y_{1}}{2})[/tex]
[tex]M = (\frac{6+3}{2} ,\frac{0+3}{2})[/tex]
[tex]M = (\frac{9}{2} ,\frac{3}{2})[/tex]
[tex]M = (4.5 ,1.5)[/tex]
Find the slope:
[tex]m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}[/tex]
[tex]m = \frac{0- 3}{6 - 3}[/tex]
[tex]m = \frac{-3}{3}[/tex]
[tex]m = \frac{-1}{1}[/tex]
Find its negative reciprocal
⊥m = 1
Substitute slope and the point into the equation of a line
y = mx + b
1.5 = 1(4.5) + b
1.5 = 4.5 + b
b = -3
⊥BC : y = x - 3
Substitute the orthocentre (3,0)
y = x - 3
0 = 3 - 3
0 = 0
LS = RS
Therefore the orthocentre is (3, 0)
