Question

What is the orthocenter of the triangle with two altitudes given by the lines x = -1 and y = x+1

Answer 1

@MaxwellFish

Answer 2

1-Find the equations of 2 segments of the triangle (for our example we will find the equations for AB, and BC)
2-Once you have the equations from step #1, you can find the slope of the corresponding perpendicular lines.
3-You will use the slopes you have found from step #2, and the corresponding opposite vertex to find the equations of the 2 lines.
4-Once you have the equation of the 2 lines from step #3, you can solve the corresponding x and y, which is the coordinates of the orthocenter.
5- Find equations of the line segments AB and BC.
To find any line segment, you will need to find the slope of the line and then the corresponding y-intercept.
A (3, 1) B(2, 2) C (3, 5)
Slope of AB = (1-2)/(3-2) = -1/1 = -1
y = mx + b (substitute m = -1, x = 3, y = 1)
1 = -1(3) + b
b = 4
Equation of AB: y = -1x + 4

Slope of BC = (2-5)/(2-3) = -3/-1 = 3
y = mx + b (substitute m = 3, x = 2, y = 2)
2 = 3(2) + b
b = -4
Equation of BC: y = 3x - 4

6- Find the slope of the corresponding perpendicular lines

Slope of AB = -1
Slope of perpendicular line to AB: -1*m = -1 -> m = 1

Slope of BC = 3
Slope of perpendicular line to BC: 3*m = -1 -> m = -1/3

7- Find the equation of the perpendicular lines

Slope of perpendicular line to AB: m = 1
We will use the coordinate of the opposite vertex (point C) to find the equation of the line.

y = mx + b (substitute m = 1, x = 3, y = 5)
5 = 1(3) + b
b = 2
Equation of perpendicular line to AB: y = 1x + 2

Slope of perpendicular line to BC: m = -1/3
We will use the coordinate of the opposite vertex (point A) to find the equation of the line.

y = mx + b (substitute m = -1/3, x = 3, y = 1)
1 = -1/3*(3) + b
b = 2
Equation of perpendicular line to AB: y = -1/3x + 2

8- solve 2 perpendicular lines

equation 1: y = 1x + 2
equation 2: y = -1/3x + 2

Solving for x and y:

1x + 2 = -1/3x + 2
4/3x = 0
x = 0

y = 1(0) + 2
y = 2

The coordinates are (0, 2). This is the orthocenter.

Answer 3

my teacher gave me that

Answer 4

hope that helps

Answer 5

omg thank you so much!!

Answer 6

No Problem