Find the coordinates of the orthocenter of the triangle with the given vertices.
L (8,0) M (10,8) N (14,0)
Well, the orthocenter of a triangle is where the altitudes of the triangle intersect. So, to find the orthocenter, we need to find the altitudes first.
Let's start by finding the slope of the line segment LM: mLM = (8 - 0) / (10 - 8) = 8/2 = 4.
The slope of the altitude passing through N will be the negative reciprocal of the slope of LM, which is -1/4.
Now, we can use the point-slope form of a line to find the equation of the altitude passing through N. Using point N (14,0), we have:
y - 0 = (-1/4)(x - 14)
y = (-1/4)x + 7/2
Similarly, we can find the equation for the altitude passing through M. The slope of MN is mMN = (8 - 0) / (10 - 14) = 8/-4 = -2.
The slope of the altitude passing through L will be the negative reciprocal of the slope of MN, which is 1/2.
Using point L (8,0), we have:
y - 0 = (1/2)(x - 8)
y = (1/2)x - 4
Now, let's solve these two equations to find the coordinates of the orthocenter.
(-1/4)x + 7/2 = (1/2)x - 4
Multiplying both sides by 4 to eliminate the fractions:
- x + 14 = 2x - 16
Adding x and 16 to both sides:
3x = 30
Dividing by 3:
x = 10
Substituting the value of x into one of the equations, let's use y = (1/2)x - 4:
y = (1/2)(10) - 4
y = 5 - 4
y = 1
So, the coordinates of the orthocenter of triangle LMN are (10, 1).
To find the coordinates of the orthocenter of a triangle, we need to find the intersection point of the altitudes.
Step 1: Find the slopes of the triangle's sides.
The slope of LM can be calculated as (y2 - y1) / (x2 - x1) = (8 - 0) / (10 - 8) = 8/2 = 4.
The slope of LN can be calculated as (y2 - y1) / (x2 - x1) = (0 - 0) / (14 - 8) = 0/6 = 0.
The slope of MN can be calculated as (y2 - y1) / (x2 - x1) = (0 - 8) / (14 - 10) = -8/4 = -2.
Step 2: Find the slopes of the altitudes.
The altitude from L to side MN will be perpendicular to MN and can be calculated as the negative reciprocal of the slope.
The slope of the altitude from L to MN will be -1/(-2) = 1/2.
Step 3: Find the equation of the line passing through L with slope 1/2.
Using the point-slope form of a linear equation, the equation of the line passing through L can be written as:
y - 0 = (1/2)(x - 8).
Simplifying, we get y = (1/2)x - 4.
Step 4: Find the coordinates of the intersection point of the line from Step 3 and the line MN.
Setting the equations of the two lines equal to each other, we have:
(1/2)x - 4 = -2x + 8.
Simplifying, we get:
(1/2)x + 2x = 8 + 4,
(5/2)x = 12,
x = 12 * 2/5 = 24/5 = 4.8.
Substituting x = 4.8 into the equation of the line, we can find y:
y = (1/2)(4.8) - 4 = 2.4 - 4 = -1.6.
Therefore, the coordinates of the orthocenter of the triangle are (4.8, -1.6).
To find the coordinates of the orthocenter of a triangle, we need to follow these steps:
1. Find the slopes of two sides of the triangle.
2. Find the slopes of the altitudes of these sides.
3. Use the slope-intercept form of the equation (y = mx + b) to find the equations of altitudes.
4. Solve the system of equations formed by the two altitude equations to find the point of intersection, which is the orthocenter.
Let's calculate step by step:
Step 1: Find the slopes of two sides of the triangle.
The slope of a line passing through two points (x1, y1) and (x2, y2) can be calculated using the formula: slope = (y2-y1)/(x2-x1).
The slopes of the sides LM, MN, and NL can be calculated as follows:
Slope of LM = (y2-y1)/(x2-x1) = (8-0)/(10-8) = 8/2 = 4
Slope of MN = (y2-y1)/(x2-x1) = (0-8)/(14-10) = -8/4 = -2
Slope of NL = (y2-y1)/(x2-x1) = (0-0)/(14-8) = 0/6 = 0
Step 2: Find the slopes of the altitudes of these sides.
The slope of an altitude is the negative reciprocal of the slope of the side it is perpendicular to.
The slopes of the altitudes can be calculated as follows:
Slope of altitude to side LM = -1/slope of LM = -1/4
Slope of altitude to side MN = -1/slope of MN = -1/(-2) = 1/2
Slope of altitude to side NL = -1/slope of NL = -1/0 (Note: the slope of NL is 0, which means it is a vertical line, and its altitude will be a horizontal line.)
Step 3: Use the slope-intercept form of the equation y = mx + b to find the equations of altitudes.
We will use the point-slope form of the equation with the slope and the coordinates of the midpoints of the sides to find the equations of the altitudes.
To find the equation of the altitude to side LM, we will use the midpoint of LM, which is ((8+10)/2, (0+8)/2) = (9, 4):
y - y1 = m(x - x1)
y - 4 = (1/2)(x - 9)
To find the equation of the altitude to side MN, we will use the midpoint of MN, which is ((10+14)/2, (8+0)/2) = (12, 4):
y - y1 = m(x - x1)
y - 4 = (1/2)(x - 12)
To find the equation of the altitude to side NL, we will use the midpoint of NL, which is ((14+8)/2, (0+0)/2) = (11, 0):
y - y1 = m(x - x1)
y - 0 = (0)(x - 11)
y - 0 = 0
y = 0 (This equation is a horizontal line passing through (11, 0)).
Step 4: Solve the system of equations formed by the two altitude equations to find the point of intersection, which is the orthocenter.
We need to solve the equations:
1) y - 4 = (1/2)(x - 9)
2) y - 4 = (1/2)(x - 12)
Solving these equations, we get x = 12 and y = 4.
Therefore, the coordinates of the orthocenter of triangle LMN are (12, 4).
The orthocentre is the intersection of the altitudes of a triangle.
You will need the equation of two of these and find their intersection
Make a sketch:
notice that LN lies on the x-axis, so the altitude from M is simply x = 10
that was too easy.
let's find the altitude to MN
slope of MN = (8-0)/(10-14) = -2
so slope of altitude is 1/2
and passing through (8,0)
0 = (1/2)(8) + b
b = -4
thus y = (1/2)x - 4
but x = 10
then y = (1/2)(10) - 4 = 1
the orthocentre is (10,1)
As an additional check, find the remaining altitude and verify by subbing in (10,1)