The orthocenter of the triangles ABC are (2, 0) and (3, 3)
<h3>How to determine the orthocenter?</h3>
<u>Question 8</u>
The coordinates are given as:
A(2,0); B(2, 4); C(6,0)
Calculate the slope of BC using:
m = (y2 - y1)/(x2 - x1)
So, we have:
m =(0 - 4)/(6 - 2)
m =-1
The perpendicular slope is:
n = -1/m
n = -1/-1
n = 1
The equation of the perpendicular line is
y = n(x - x1) + y1
Where:
(x1, y1) = (2,0)
So, we have:
y = 1(x - 2) + 0
y = x - 2
Calculate the slope of AC using:
m = (y2 - y1)/(x2 - x1)
So, we have:
m =(0 - 0)/(6 - 2)
m =0
The perpendicular slope is:
n = -1/m
n = undefined
This means that the perpendicular line is a vertical line
The equation of the perpendicular line is
x = x1
Where:
(x1, y1) = (2,4)
So, we have:
x = 2
Substitute x = 2 in y = x - 2
y = 2 - 2
y = 0
So, we have:
(x, y) = (2, 0)
Hence, the orthocenter of ABC is (2, 0)
<u>Question 9</u>
The coordinates are given as:
A(1,1); B(3, 4); C(6,1)
Calculate the slope of BC using:
m = (y2 - y1)/(x2 - x1)
So, we have:
m =(1 - 4)/(6 - 3)
m =-1
The perpendicular slope is:
n = -1/m
n = -1/-1
n = 1
The equation of the perpendicular line is
y = n(x - x1) + y1
Where:
(x1, y1) = (1,1)
So, we have:
y = 1(x - 1) + 1
y = x
Calculate the slope of AC using:
m = (y2 - y1)/(x2 - x1)
So, we have:
m =(1 - 1)/(6 - 1)
m =0
The perpendicular slope is:
n = -1/m
n = undefined
This means that the perpendicular line is a vertical line
The equation of the perpendicular line is
x = x1
Where:
(x1, y1) = (3,4)
So, we have:
x = 3
Substitute x = 3 in y = x
y = 3
So, we have:
(x, y) = (3, 3)
Hence, the orthocenter of ABC is (3, 3)
<h3>How to solve for x?</h3>
<u>Question 18 - 20</u>
Line 2x and x+ 3 are congruent lines
So, we have:
2x = x +3
Solve for x
x= 3
Line 3x and 2x + 5 are congruent lines
So, we have:
3x = 2x +5
Solve for x
x = 5
Line 2x - 3 and x + 2 are congruent lines
So, we have:
2x - 3 = x + 2
Solve for x
x = 5
<u>Question 21 - 23</u>
Line x + 5 and 3x + 7 are congruent lines
So, we have:
x + 5 = 3x + 7
Solve for x
x = -1
Line x + 6 and 5x - 4 are congruent lines
So, we have:
x + 6 = 5x - 4
Solve for x
x = 2.5
Line x + 8 and 5x + 7 are congruent lines
So, we have:
x + 8 = 5x + 7
Solve for x
x = 1/4
Read more about orthocenter at:
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