Triangle POB has vertices:
P(0 , 4)
O(-3 , 0)
B(3, 0).
Determine the orthocentre of the triangle.
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puipoupoip
the differens of the squares of two consecutive odd integers is 24 find the integres
To find the orthocenter of a triangle, you need to find the intersection point of its altitudes. Recall that an altitude of a triangle is a line segment drawn from a vertex perpendicular to the opposite side.
Let's find the equation of the three altitudes and then solve for their intersection point, which will be the orthocenter.
Step 1: Find the equation of the line passing through points P and O.
The slope of the line passing through P(0, 4) and O(-3, 0) can be found using the formula:
m = (y2 - y1) / (x2 - x1)
m = (0 - 4) / (-3 - 0)
m = -4 / -3
m = 4/3
Using the point-slope form of a line equation, the equation of the line passing through P and O is:
y - y1 = m(x - x1)
y - 4 = (4/3)(x - 0)
y - 4 = (4/3)x
Simplifying the equation:
3y - 12 = 4x
Step 2: Find the equation of the line passing through points P and B.
The slope of the line passing through P(0, 4) and B(3, 0) can be found using the formula:
m = (y2 - y1) / (x2 - x1)
m = (0 - 4) / (3 - 0)
m = -4 / 3
Using the point-slope form of a line equation, the equation of the line passing through P and B is:
y - y1 = m(x - x1)
y - 4 = (-4/3)(x - 0)
y - 4 = (-4/3)x
Simplifying the equation:
3y - 12 = -4x
Step 3: Find the equation of the line passing through points O and B.
Since the line segments OB and PO are horizontal and perpendicular to each other, their equations will be of the form y = constant.
The equation of the line passing through O(-3, 0) and B(3, 0) is:
y = 0
Step 4: Find the intersection point of the altitudes (orthocenter).
To find the orthocenter, we need to find the point of intersection of the three lines. This can be done by solving the system of equations formed by the three line equations.
Solving the system of equations:
3y - 12 = 4x
3y - 12 = -4x
y = 0
Substituting the third equation into the first two equations:
3(0) - 12 = 4x
3(0) - 12 = -4x
Simplifying:
-12 = 4x
12 = -4x
Dividing both sides of the equation by 4:
x = -3
Substituting the value of x back into any of the three line equations to find y:
3(0) - 12 = 4(-3)
-12 = -12
The orthocenter of triangle POB is at the point (-3, 0).
Therefore, the orthocenter of triangle POB is the point (-3, 0).