Finding the Equation of the Line Containing the Side Opposite to Vertex A

Oct 13, 2025 by talynt.org 67 views

Introduction

In this problem, we are given the vertices of a triangle as A(0,1), B(1,3), and C(4,-2). We are asked to determine the equation and length of the altitude drawn from vertex A to the opposite side of the triangle. To solve this problem, we will first find the equation of the line containing the side opposite to vertex A, and then find the equation of the altitude. We will also calculate the length of the altitude.

Finding the Equation of the Line Containing the Side Opposite to Vertex A

The side opposite to vertex A is BC. To find the equation of the line containing BC, we need to find the slope of the line. The slope of the line passing through two points (x1, y1) and (x2, y2) is given by the formula:

m = (y2 - y1) / (x2 - x1)

In this case, the two points are B(1,3) and C(4,-2). Plugging these values into the formula, we get:

m = (-2 - 3) / (4 - 1) m = -5 / 3 m = -5/3

Now that we have the slope, we can use the point-slope form of the equation of a line to find the equation of the line containing BC. The point-slope form is given by:

y - y1 = m(x - x1)

We can use point B(1,3) as the point (x1, y1). Plugging in the values, we get:

y - 3 = (-5/3)(x - 1)

To simplify the equation, we can multiply both sides by 3 to get rid of the fraction:

3(y - 3) = -5(x - 1) 3y - 9 = -5x + 5

Now, we can rearrange the equation to get it in the standard form:

5x + 3y - 14 = 0

Finding the Equation of the Altitude

The altitude is a line that passes through vertex A and is perpendicular to the line containing BC. To find the equation of the altitude, we need to find the slope of the altitude. Since the altitude is perpendicular to the line containing BC, the slope of the altitude is the negative reciprocal of the slope of the line containing BC. The slope of the line containing BC is -5/3, so the slope of the altitude is:

m_altitude = -1 / (-5/3) m_altitude = 3/5

Now that we have the slope of the altitude, we can use the point-slope form of the equation of a line to find the equation of the altitude. We can use point A(0,1) as the point (x1, y1). Plugging in the values, we get:

y - 1 = (3/5)(x - 0)

To simplify the equation, we can multiply both sides by 5 to get rid of the fraction:

5(y - 1) = 3x 5y - 5 = 3x

Now, we can rearrange the equation to get it in the standard form:

3x - 5y + 5 = 0

Calculating the Length of the Altitude

To calculate the length of the altitude, we need to find the distance between the point of intersection of the altitude and the line containing BC, and vertex A. We can find the point of intersection by solving the system of equations formed by the altitude and the line containing BC.

The equation of the line containing BC is:

5x + 3y - 14 = 0

The equation of the altitude is:

3x - 5y + 5 = 0

We can solve this system of equations by multiplying the first equation by 5 and the second equation by 3, and then adding the two equations to eliminate the y-variable:

25x + 15y - 70 = 0 9x - 15y + 15 = 0

Adding the two equations, we get:

34x - 55 = 0

Solving for x, we get:

x = 55/34

Now that we have the value of x, we can substitute it into one of the original equations to find the value of y. We can use the equation of the line containing BC:

5x + 3y - 14 = 0

Substituting x = 55/34, we get:

5(55/34) + 3y - 14 = 0

Solving for y, we get:

y = (14 - 275/34) / 3 y = (476 - 275) / 102 y = 201 / 102 y = 67/34

Now that we have the values of x and y, we can find the distance between the point of intersection and vertex A using the distance formula:

d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

In this case, the two points are (55/34, 67/34) and (0,1). Plugging these values into the formula, we get:

d = sqrt(((55/34) - 0)^2 + ((67/34) - 1)^2) d = sqrt((55/34)^2 + (33/34)^2) d = sqrt(3025/1156 + 1089/1156) d = sqrt(4114/1156) d = sqrt(3.56)

d ≈ 1.88

Therefore, the length of the altitude is approximately 1.88 units.

Conclusion

In this problem, we determined the equation and length of the altitude drawn from vertex A to the opposite side of the triangle. We found the equation of the line containing the side opposite to vertex A, and then found the equation of the altitude. We also calculated the length of the altitude by finding the distance between the point of intersection of the altitude and the line containing BC, and vertex A.

b) Finding the Equations of Two Lines Through the Point (2, -3) with Slopes 2 and -1/2

Introduction

In this problem, we are given the point (2, -3) and two slopes, 2 and -1/2. We are asked to find the equations of two lines that pass through the point (2, -3) and have the given slopes.

Finding the Equation of the Line with Slope 2

The point-slope form of the equation of a line is given by:

y - y1 = m(x - x1)

We can use point (2, -3) as the point (x1, y1). Plugging in the values, we get:

y - (-3) = 2(x - 2)

To simplify the equation, we can multiply both sides by 1 to get rid of the fraction:

y + 3 = 2(x - 2) y + 3 = 2x - 4

Now, we can rearrange the equation to get it in the standard form:

2x - y - 7 = 0

Finding the Equation of the Line with Slope -1/2

The point-slope form of the equation of a line is given by:

y - y1 = m(x - x1)

We can use point (2, -3) as the point (x1, y1). Plugging in the values, we get:

y - (-3) = (-1/2)(x - 2)

To simplify the equation, we can multiply both sides by 2 to get rid of the fraction:

2(y - (-3)) = -1(x - 2) 2y + 6 = -x + 2

Now, we can rearrange the equation to get it in the standard form:

x + 2y + 4 = 0

Conclusion

In this problem, we found the equations of two lines that pass through the point (2, -3) and have the given slopes. We used the point-slope form of the equation of a line to find the equations of the two lines.

Introduction

In this article, we will answer some frequently asked questions about determining the equation and length of the altitude drawn from vertex A to the opposite side of the triangle. We will cover topics such as finding the equation of the line containing the side opposite to vertex A, finding the equation of the altitude, and calculating the length of the altitude.

Q: What is the equation of the line containing the side opposite to vertex A?

A: The equation of the line containing the side opposite to vertex A is 5x + 3y - 14 = 0.

Q: How do I find the equation of the altitude?

A: To find the equation of the altitude, you need to find the slope of the altitude, which is the negative reciprocal of the slope of the line containing the side opposite to vertex A. Then, you can use the point-slope form of the equation of a line to find the equation of the altitude.

Q: What is the length of the altitude?

A: The length of the altitude is approximately 1.88 units.

Q: How do I calculate the length of the altitude?

A: To calculate the length of the altitude, you need to find the distance between the point of intersection of the altitude and the line containing the side opposite to vertex A, and vertex A. You can use the distance formula to find the length of the altitude.

Q: What is the point of intersection of the altitude and the line containing the side opposite to vertex A?

A: The point of intersection of the altitude and the line containing the side opposite to vertex A is (55/34, 67/34).

Q: How do I find the equation of a line that passes through a given point and has a given slope?

A: To find the equation of a line that passes through a given point and has a given slope, you can use the point-slope form of the equation of a line. The point-slope form is given by:

y - y1 = m(x - x1)

where (x1, y1) is the given point and m is the given slope.

Q: What is the equation of the line with slope 2 that passes through the point (2, -3)?

A: The equation of the line with slope 2 that passes through the point (2, -3) is 2x - y - 7 = 0.

Q: What is the equation of the line with slope -1/2 that passes through the point (2, -3)?

A: The equation of the line with slope -1/2 that passes through the point (2, -3) is x + 2y + 4 = 0.

Conclusion

In this article, we answered some frequently asked questions about determining the equation and length of the altitude drawn from vertex A to the opposite side of the triangle. We covered topics such as finding the equation of the line containing the side opposite to vertex A, finding the equation of the altitude, and calculating the length of the altitude.

Additional Resources

For more information on finding the equation of a line, see our article on "Finding the Equation of a Line".
For more information on calculating the length of a line segment, see our article on "Calculating the Length of a Line Segment".
For more information on the point-slope form of the equation of a line, see our article on "The Point-Slope Form of the Equation of a Line".