The Point C Has Coordinates (2,-3) And The Point D Has Coordinates (4,6). Find The Equation Of The Line Perpendicular To CD And Passing Through D.

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Introduction

In mathematics, finding the equation of a line perpendicular to a given line and passing through a specific point is a fundamental problem. This problem involves understanding the concept of slope and using it to determine the equation of the perpendicular line. In this article, we will discuss how to find the equation of a line perpendicular to CD and passing through D, given the coordinates of points C and D.

Understanding Slope

The slope of a line is a measure of how steep it is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. The slope of a line can be represented by the formula:

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

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

Finding the Slope of CD

To find the slope of CD, we can use the coordinates of points C (2, -3) and D (4, 6). Plugging these values into the formula, we get:

m = (6 - (-3)) / (4 - 2) m = (6 + 3) / 2 m = 9 / 2 m = 4.5

So, the slope of CD is 4.5.

Finding the Slope of the Perpendicular Line

The slope of a line perpendicular to CD is the negative reciprocal of the slope of CD. This means that if the slope of CD is 4.5, the slope of the perpendicular line is -1/4.5.

Finding the Equation of the Perpendicular Line

To find the equation of the perpendicular line, we need to use the point-slope form of a line, which is given by:

y - y1 = m(x - x1)

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

In this case, we know that the perpendicular line passes through point D (4, 6) and has a slope of -1/4.5. Plugging these values into the equation, we get:

y - 6 = (-1/4.5)(x - 4)

To simplify this equation, we can multiply both sides by 4.5 to get rid of the fraction:

4.5(y - 6) = -1(x - 4) 4.5y - 27 = -x + 4

Now, we can rearrange this equation to get it in the standard form of a line, which is:

ax + by = c

where a, b, and c are constants.

4.5y = x - 31 4.5y = x - 31 x - 4.5y = 31

So, the equation of the line perpendicular to CD and passing through D is x - 4.5y = 31.

Conclusion

In this article, we discussed how to find the equation of a line perpendicular to CD and passing through D, given the coordinates of points C and D. We used the concept of slope and the point-slope form of a line to determine the equation of the perpendicular line. The equation of the line perpendicular to CD and passing through D is x - 4.5y = 31.

Example Problems

  1. Find the equation of the line perpendicular to the line passing through points A (1, 2) and B (3, 4).
  2. Find the equation of the line perpendicular to the line passing through points C (2, 3) and D (4, 5).
  3. Find the equation of the line perpendicular to the line passing through points E (1, 1) and F (2, 2).

Solutions

  1. To find the equation of the line perpendicular to the line passing through points A (1, 2) and B (3, 4), we need to find the slope of the line passing through these points. The slope of the line passing through points A and B is:

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

The slope of the line perpendicular to the line passing through points A and B is the negative reciprocal of the slope of the line passing through these points. Therefore, the slope of the line perpendicular to the line passing through points A and B is -1.

To find the equation of the line perpendicular to the line passing through points A and B, we can use the point-slope form of a line. We know that the line passes through point A (1, 2) and has a slope of -1. Plugging these values into the equation, we get:

y - 2 = -1(x - 1)

To simplify this equation, we can multiply both sides by -1 to get rid of the negative sign:

y - 2 = x - 1 y = x - 1 + 2 y = x + 1

So, the equation of the line perpendicular to the line passing through points A and B is y = x + 1.

  1. To find the equation of the line perpendicular to the line passing through points C (2, 3) and D (4, 5), we need to find the slope of the line passing through these points. The slope of the line passing through points C and D is:

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

The slope of the line perpendicular to the line passing through points C and D is the negative reciprocal of the slope of the line passing through these points. Therefore, the slope of the line perpendicular to the line passing through points C and D is -1.

To find the equation of the line perpendicular to the line passing through points C and D, we can use the point-slope form of a line. We know that the line passes through point D (4, 5) and has a slope of -1. Plugging these values into the equation, we get:

y - 5 = -1(x - 4)

To simplify this equation, we can multiply both sides by -1 to get rid of the negative sign:

y - 5 = x - 4 y = x - 4 + 5 y = x + 1

So, the equation of the line perpendicular to the line passing through points C and D is y = x + 1.

  1. To find the equation of the line perpendicular to the line passing through points E (1, 1) and F (2, 2), we need to find the slope of the line passing through these points. The slope of the line passing through points E and F is:

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

The slope of the line perpendicular to the line passing through points E and F is the negative reciprocal of the slope of the line passing through these points. Therefore, the slope of the line perpendicular to the line passing through points E and F is -1.

To find the equation of the line perpendicular to the line passing through points E and F, we can use the point-slope form of a line. We know that the line passes through point F (2, 2) and has a slope of -1. Plugging these values into the equation, we get:

y - 2 = -1(x - 2)

To simplify this equation, we can multiply both sides by -1 to get rid of the negative sign:

y - 2 = x - 2 y = x - 2 + 2 y = x

Q: What is the equation of the line perpendicular to CD and passing through D?

A: The equation of the line perpendicular to CD and passing through D is x - 4.5y = 31.

Q: How do I find the equation of a line perpendicular to a given line and passing through a specific point?

A: To find the equation of a line perpendicular to a given line and passing through a specific point, you need to follow these steps:

  1. Find the slope of the given line using the coordinates of two points on the line.
  2. Find the slope of the perpendicular line by taking the negative reciprocal of the slope of the given line.
  3. Use the point-slope form of a line to find the equation of the perpendicular line.

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

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

y - y1 = m(x - x1)

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

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 negative reciprocal of a slope?

A: The negative reciprocal of a slope is the slope that is perpendicular to the original slope. If the original slope is m, then the negative reciprocal of m is -1/m.

Q: Can I use the slope-intercept form of a line to find the equation of a line perpendicular to a given line and passing through a specific point?

A: Yes, you can use the slope-intercept form of a line to find the equation of a line perpendicular to a given line and passing through a specific point. The slope-intercept form of a line is given by:

y = mx + b

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

To find the equation of a line perpendicular to a given line and passing through a specific point, you can use the following steps:

  1. Find the slope of the given line using the coordinates of two points on the line.
  2. Find the slope of the perpendicular line by taking the negative reciprocal of the slope of the given line.
  3. Use the slope-intercept form of a line to find the equation of the perpendicular line.

Q: How do I find the equation of a line perpendicular to a given line and passing through a specific point using the slope-intercept form of a line?

A: To find the equation of a line perpendicular to a given line and passing through a specific point using the slope-intercept form of a line, you can follow these steps:

  1. Find the slope of the given line using the coordinates of two points on the line.
  2. Find the slope of the perpendicular line by taking the negative reciprocal of the slope of the given line.
  3. Use the slope-intercept form of a line to find the equation of the perpendicular line.

For example, if the given line has a slope of 2 and passes through the point (1, 3), and you want to find the equation of a line perpendicular to the given line and passing through the point (2, 4), you can use the following steps:

  1. Find the slope of the given line: m = 2
  2. Find the slope of the perpendicular line: m = -1/2
  3. Use the slope-intercept form of a line to find the equation of the perpendicular line: y = -1/2x + b

To find the value of b, you can substitute the coordinates of the point (2, 4) into the equation:

4 = -1/2(2) + b 4 = -1 + b b = 5

Therefore, the equation of the line perpendicular to the given line and passing through the point (2, 4) is:

y = -1/2x + 5

Q: Can I use a graphing calculator to find the equation of a line perpendicular to a given line and passing through a specific point?

A: Yes, you can use a graphing calculator to find the equation of a line perpendicular to a given line and passing through a specific point. You can use the calculator to graph the given line and the point, and then use the calculator's built-in functions to find the equation of the perpendicular line.

Q: How do I use a graphing calculator to find the equation of a line perpendicular to a given line and passing through a specific point?

A: To use a graphing calculator to find the equation of a line perpendicular to a given line and passing through a specific point, you can follow these steps:

  1. Graph the given line and the point on the calculator.
  2. Use the calculator's built-in functions to find the slope of the given line.
  3. Use the calculator's built-in functions to find the slope of the perpendicular line.
  4. Use the calculator's built-in functions to find the equation of the perpendicular line.

For example, if the given line has a slope of 2 and passes through the point (1, 3), and you want to find the equation of a line perpendicular to the given line and passing through the point (2, 4), you can use the following steps:

  1. Graph the given line and the point on the calculator.
  2. Use the calculator's built-in functions to find the slope of the given line: m = 2
  3. Use the calculator's built-in functions to find the slope of the perpendicular line: m = -1/2
  4. Use the calculator's built-in functions to find the equation of the perpendicular line: y = -1/2x + b

To find the value of b, you can substitute the coordinates of the point (2, 4) into the equation:

4 = -1/2(2) + b 4 = -1 + b b = 5

Therefore, the equation of the line perpendicular to the given line and passing through the point (2, 4) is:

y = -1/2x + 5