A Triangle Has Vertices A(2,5) B(1,2) C(-5,1).Determine The Equation Of Line BC

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Introduction

In geometry, a line is a set of points that extend infinitely in two directions. To determine the equation of a line, we need to know two points on the line. In this article, we will use the coordinates of points B and C to find the equation of line BC.

What is the Equation of a Line?

The equation of a line is a mathematical expression that describes the relationship between the x and y coordinates of any point on the line. The general form of the equation of a line is:

y = mx + b

where m is the slope of the line and b is the y-intercept.

Finding the Slope of Line BC

To find the slope of line BC, we need to use the coordinates of points B and C. The slope of a line is defined as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.

The coordinates of points B and C are:

B(1,2) C(-5,1)

To find the slope, we can use the formula:

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

where (x1, y1) and (x2, y2) are the coordinates of points B and C, respectively.

Plugging in the values, we get:

m = (1 - 2) / (-5 - 1) m = -1 / -6 m = 1/6

Finding the Equation of Line BC

Now that we have the slope of line BC, we can use it to find the equation of the line. We can use the point-slope form of the equation of a line, which is:

y - y1 = m(x - x1)

where (x1, y1) is a point on the line and m is the slope.

We can use point B(1,2) as the point on the line. Plugging in the values, we get:

y - 2 = (1/6)(x - 1)

To simplify the equation, we can multiply both sides by 6 to get rid of the fraction:

6(y - 2) = x - 1

Expanding the left side, we get:

6y - 12 = x - 1

Adding 12 to both sides, we get:

6y = x + 11

Dividing both sides by 6, we get:

y = (1/6)x + 11/6

Conclusion

In this article, we used the coordinates of points B and C to find the equation of line BC. We first found the slope of the line using the coordinates of the two points, and then used the point-slope form of the equation of a line to find the equation of the line. The equation of line BC is y = (1/6)x + 11/6.

Example Problems

  • Find the equation of line AC using the coordinates of points A(2,5) and C(-5,1).
  • Find the equation of line AB using the coordinates of points A(2,5) and B(1,2).

Step-by-Step Solution

To find the equation of line AC, we need to follow the same steps as before:

  1. Find the slope of line AC using the coordinates of points A and C.
  2. Use the point-slope form of the equation of a line to find the equation of the line.

To find the slope of line AC, we can use the formula:

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

where (x1, y1) and (x2, y2) are the coordinates of points A and C, respectively.

Plugging in the values, we get:

m = (1 - 5) / (-5 - 2) m = -4 / -7 m = 4/7

Now that we have the slope of line AC, we can use it to find the equation of the line. We can use point A(2,5) as the point on the line. Plugging in the values, we get:

y - 5 = (4/7)(x - 2)

To simplify the equation, we can multiply both sides by 7 to get rid of the fraction:

7(y - 5) = 4(x - 2)

Expanding the left side, we get:

7y - 35 = 4x - 8

Adding 35 to both sides, we get:

7y = 4x + 27

Dividing both sides by 7, we get:

y = (4/7)x + 27/7

The equation of line AC is y = (4/7)x + 27/7.

Frequently Asked Questions

  • What is the equation of line BC?
  • How do I find the equation of a line using the coordinates of two points?
  • What is the slope of line BC?

Conclusion

In this article, we used the coordinates of points B and C to find the equation of line BC. We first found the slope of the line using the coordinates of the two points, and then used the point-slope form of the equation of a line to find the equation of the line. The equation of line BC is y = (1/6)x + 11/6. We also found the equation of line AC using the coordinates of points A and C. The equation of line AC is y = (4/7)x + 27/7.

Introduction

In our previous article, we used the coordinates of points B and C to find the equation of line BC. We first found the slope of the line using the coordinates of the two points, and then used the point-slope form of the equation of a line to find the equation of the line. The equation of line BC is y = (1/6)x + 11/6. In this article, we will answer some frequently asked questions about the equation of line BC.

Q&A

Q: What is the equation of line BC?

A: The equation of line BC is y = (1/6)x + 11/6.

Q: How do I find the equation of a line using the coordinates of two points?

A: To find the equation of a line using the coordinates of two points, you need to follow these steps:

  1. Find the slope of the line using the coordinates of the two points.
  2. Use the point-slope form of the equation of a line to find the equation of the line.

Q: What is the slope of line BC?

A: The slope of line BC is 1/6.

Q: How do I find the slope of a line using the coordinates of two points?

A: To find the slope of a line using the coordinates of two points, you can use the formula:

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

where (x1, y1) and (x2, y2) are the coordinates of the two points.

Q: What is the point-slope form of the equation of a line?

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

y - y1 = m(x - x1)

where (x1, y1) is a point on the line and m is the slope.

Q: How do I use the point-slope form of the equation of a line to find the equation of a line?

A: To use the point-slope form of the equation of a line to find the equation of a line, you need to follow these steps:

  1. Plug in the values of the point and the slope into the equation.
  2. Simplify the equation to get the final equation of the line.

Q: What is the equation of line AC?

A: The equation of line AC is y = (4/7)x + 27/7.

Q: How do I find the equation of line AC?

A: To find the equation of line AC, you need to follow the same steps as before:

  1. Find the slope of line AC using the coordinates of points A and C.
  2. Use the point-slope form of the equation of a line to find the equation of the line.

Example Problems

  • Find the equation of line AB using the coordinates of points A(2,5) and B(1,2).
  • Find the equation of line CD using the coordinates of points C(-5,1) and D(3,4).

Step-by-Step Solution

To find the equation of line AB, you need to follow the same steps as before:

  1. Find the slope of line AB using the coordinates of points A and B.
  2. Use the point-slope form of the equation of a line to find the equation of the line.

To find the slope of line AB, you can use the formula:

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

where (x1, y1) and (x2, y2) are the coordinates of points A and B, respectively.

Plugging in the values, you get:

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

Now that you have the slope of line AB, you can use it to find the equation of the line. You can use point A(2,5) as the point on the line. Plugging in the values, you get:

y - 5 = 3(x - 2)

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

y - 5 = 3x - 6

Adding 5 to both sides, you get:

y = 3x - 1

The equation of line AB is y = 3x - 1.

Frequently Asked Questions

  • What is the equation of line BC?
  • How do I find the equation of a line using the coordinates of two points?
  • What is the slope of line BC?

Conclusion

In this article, we answered some frequently asked questions about the equation of line BC. We also found the equation of line AB using the coordinates of points A and B. The equation of line AB is y = 3x - 1. We hope that this article has been helpful in answering your questions about the equation of line BC.