Geometry Problem: Finding Equations & Properties In A Triangle
Hey guys! Let's dive into a fun geometry problem involving triangles, coordinates, and equations. This problem focuses on finding different equations and exploring properties within a triangle. So, grab your thinking caps, and let's get started!
Setting the Stage
We're working in a plane that's set up with an orthonormal coordinate system (O, i, j). Think of it as your standard graph paper setup, where the axes are perpendicular and the units are the same on both axes. We've got two points marked on this plane: L at (-3, 2) and K at (3, 4). Now, imagine we draw a triangle by connecting these points to the origin, O. This gives us triangle OLK. A crucial element in this problem is the altitude from O in triangle OLK, which we'll call do. Remember, an altitude is a line segment from a vertex (corner) of the triangle perpendicular to the opposite side. We're given that the equation of the line containing this altitude do is 3x + y = 0. This is a key piece of information that we'll use to unlock the rest of the problem.
1) Determining Equations
Our primary goal here is to find the equations of various lines associated with triangle OLK. This involves using the coordinates of the points and the given equation of the altitude to determine the equations of the sides of the triangle and other relevant lines. Let's break down the process:
Finding the Equation of Line (LK)
First, we need to find the equation of the line that passes through points L and K. To do this, we'll use the two-point form of a line equation. Remember that the two-point form allows us to find the equation of a line if we know the coordinates of two points on that line. Given points L(-3, 2) and K(3, 4), the formula looks like this:
(y - y1) / (x - x1) = (y2 - y1) / (x2 - x1)
Where (x1, y1) and (x2, y2) are the coordinates of our points L and K. Let's plug in the values:
(y - 2) / (x - (-3)) = (4 - 2) / (3 - (-3))
Simplifying this, we get:
(y - 2) / (x + 3) = 2 / 6
(y - 2) / (x + 3) = 1 / 3
Now, cross-multiply to get rid of the fractions:
3(y - 2) = 1(x + 3)
Expand and rearrange to get the equation in the standard form (Ax + By + C = 0):
3y - 6 = x + 3
x - 3y + 9 = 0
So, the equation of the line (LK) is x - 3y + 9 = 0. This equation is crucial because it defines one of the sides of our triangle, and we'll need it for further calculations.
Finding the Equation of the Altitude from K
Next, we'll find the equation of the altitude from point K in triangle OLK. This altitude is a line segment that extends from vertex K and is perpendicular to the opposite side, which is line (OL). To find the equation of this altitude, we need to determine its slope. Remember that perpendicular lines have slopes that are negative reciprocals of each other. So, first, let's find the slope of line (OL).
The slope of a line passing through two points (x1, y1) and (x2, y2) is given by:
m = (y2 - y1) / (x2 - x1)
For line (OL), the points are O(0, 0) and L(-3, 2). Plugging these values into the slope formula, we get:
m(OL) = (2 - 0) / (-3 - 0) = 2 / -3 = -2/3
Now, to find the slope of the altitude from K, we take the negative reciprocal of m(OL):
m(altitude from K) = -1 / (-2/3) = 3/2
We now have the slope of the altitude from K, and we know it passes through point K(3, 4). We can use the point-slope form of a line equation to find its equation:
y - y1 = m(x - x1)
Plugging in the values, we get:
y - 4 = (3/2)(x - 3)
To get rid of the fraction, multiply both sides by 2:
2(y - 4) = 3(x - 3)
Expand and rearrange to get the equation in standard form:
2y - 8 = 3x - 9
3x - 2y - 1 = 0
Therefore, the equation of the altitude from K is 3x - 2y - 1 = 0. This is another key equation that helps us understand the geometry of our triangle.
Finding the Equation of the Altitude from L
Now, let's tackle the altitude from point L in triangle OLK. This altitude is a line segment from vertex L perpendicular to the opposite side, which is line (OK). We'll follow a similar process to what we did for the altitude from K. First, we need to find the slope of line (OK).
The points for line (OK) are O(0, 0) and K(3, 4). Using the slope formula:
m(OK) = (4 - 0) / (3 - 0) = 4 / 3
To find the slope of the altitude from L, we take the negative reciprocal of m(OK):
m(altitude from L) = -1 / (4/3) = -3/4
We now have the slope of the altitude from L, and we know it passes through point L(-3, 2). Using the point-slope form of a line equation:
y - y1 = m(x - x1)
Plugging in the values, we get:
y - 2 = (-3/4)(x - (-3))
y - 2 = (-3/4)(x + 3)
To eliminate the fraction, multiply both sides by 4:
4(y - 2) = -3(x + 3)
Expand and rearrange to get the equation in standard form:
4y - 8 = -3x - 9
3x + 4y + 1 = 0
So, the equation of the altitude from L is 3x + 4y + 1 = 0. We now have the equations for all three altitudes of the triangle.
Recap of Equations Found
Let's quickly recap the equations we've found so far:
- Equation of line (LK): x - 3y + 9 = 0
- Equation of altitude from K: 3x - 2y - 1 = 0
- Equation of altitude from L: 3x + 4y + 1 = 0
These equations are essential for understanding the geometry of triangle OLK and will be used in the subsequent parts of the problem.
Determining the Coordinates
After finding the equations, the next logical step in solving this geometry problem is to determine specific coordinates. These coordinates might represent important points within the triangle, such as the intersection of altitudes or the foot of a perpendicular from a vertex to the opposite side. Let's break down how we'll approach this.
Finding the Foot of the Altitude from O
We're given that do, the altitude from O, has the equation 3x + y = 0. To find the foot of this altitude (the point where it intersects line (LK)), we need to solve the system of equations formed by the equation of do and the equation of line (LK).
We already found the equation of line (LK) to be x - 3y + 9 = 0. So, our system of equations is:
- 3x + y = 0
- x - 3y + 9 = 0
There are a couple of ways we can solve this system. One common method is substitution. From equation (1), we can easily isolate y:
y = -3x
Now, substitute this expression for y into equation (2):
x - 3(-3x) + 9 = 0
Simplify and solve for x:
x + 9x + 9 = 0
10x + 9 = 0
10x = -9
x = -9/10
Now that we have the value of x, we can substitute it back into the equation y = -3x to find the value of y:
y = -3(-9/10)
y = 27/10
Therefore, the coordinates of the foot of the altitude from O are (-9/10, 27/10). This point is crucial because it represents the point where the altitude from O perpendicularly intersects the side (LK) of the triangle.
Finding the Orthocenter
The orthocenter of a triangle is the point where all three altitudes intersect. We already have the equations for the three altitudes: altitude from O (3x + y = 0), altitude from K (3x - 2y - 1 = 0), and altitude from L (3x + 4y + 1 = 0). To find the orthocenter, we need to solve the system of equations formed by any two of these altitudes.
Let's choose the altitudes from O and K. Our system of equations is:
- 3x + y = 0
- 3x - 2y - 1 = 0
We can use either substitution or elimination to solve this system. Let's use elimination. Subtract equation (2) from equation (1):
(3x + y) - (3x - 2y - 1) = 0 - 0
3x + y - 3x + 2y + 1 = 0
3y + 1 = 0
3y = -1
y = -1/3
Now, substitute this value of y back into equation (1) to find x:
3x + (-1/3) = 0
3x = 1/3
x = 1/9
Therefore, the coordinates of the orthocenter are (1/9, -1/3). This point is a significant geometric center of the triangle, and finding it helps us understand the relationships between the altitudes and the triangle's vertices.
Wrapping Up This Section
In this section, we've successfully determined the coordinates of two important points: the foot of the altitude from O and the orthocenter of triangle OLK. These calculations involved solving systems of linear equations, a fundamental skill in geometry and algebra. With these coordinates in hand, we're well-equipped to tackle the next challenges in this problem, which might involve calculating distances, areas, or other geometric properties of the triangle. Keep up the great work, guys! We're making solid progress in understanding this geometric puzzle.
Let me know if you'd like to delve deeper into any specific part or if there's anything else I can assist you with! This is turning out to be quite the insightful journey into the world of geometry, and I'm here to help you every step of the way. Let's keep exploring and unlocking the secrets hidden within triangles and coordinate systems!