Find the equation of the line containing the point ( 1, -2 ) and perpendicular to the line containing the points ( 1, 5 ) and ( 3, 2 )

slope between (1,5) and (3,2)

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

so the slope of our needed line is 2/3

equation of new line
y+2 = (2/3)(x-1)
3y + 6 = 2x - 2

arrange it into the form you want

Why did the perpendicular line join the party? Because it couldn't resist the opportunity to be different and stand out! Now, let's find the equation.

First, let's find the slope of the given line using the formula:

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

Using the points (1, 5) and (3, 2), we get:

slope = (2 - 5) / (3 - 1) = -3 / 2 = -1.5

Since the new line is perpendicular to the given line, the slope of the new line is the negative reciprocal of -1.5. To find it, we flip the fraction and change the sign:

new slope = 2 / 3

Now, we use the point-slope form of the equation:

y - y1 = m(x - x1)

Plugging in the point (1, -2) and the slope 2/3, we get:

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

Simplifying, we have:

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

Rearranging the equation, we get:

3y + 6 = 2x - 2

And finally:

2x - 3y = 8

There you have it! The equation of the line is 2x - 3y = 8. Now go tell that perpendicular line it made the right choice by going against the grain!

To find the equation of a line perpendicular to another line, we first need to find the slope of the given line.

Step 1: Find the slope of the given line.
The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula:
slope = (y2 - y1) / (x2 - x1)

Given the points (1, 5) and (3, 2), we can find the slope of the line containing these points:
slope = (2 - 5) / (3 - 1)
slope = -3 / 2
slope = -1.5

Step 2: Find the slope of the line perpendicular to the given line.
To find the slope of a line perpendicular to another line, we use the property that the product of the slopes of two perpendicular lines is -1.
So, the slope of the line perpendicular to the given line is the negative reciprocal of the given slope:
perpendicular slope = -1 / (-1.5)
perpendicular slope = 2/3

Step 3: Use the point-slope form of the equation.

The equation of a line passing through the point (x1, y1) with slope m is given by the point-slope form:
y - y1 = m(x - x1)

Using the point (1, -2) and the perpendicular slope 2/3, we can write the equation as:
y - (-2) = (2/3)(x - 1)

Simplifying the equation:
y + 2 = (2/3)(x - 1)

Now, we can rewrite the equation in standard form:
3(y + 2) = 2(x - 1)

Expanding and rearranging:
3y + 6 = 2x - 2

Rearranging again, we get the equation of the line:
2x - 3y = 8

So, the equation of the line containing the point (1, -2) and perpendicular to the line containing the points (1, 5) and (3, 2) is 2x - 3y = 8.

To find the equation of the line perpendicular to another line, we first need to find the slope of the given line.

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).

For the given line with points (1, 5) and (3, 2), the slope is:

slope = (2 - 5) / (3 - 1) = -3 / 2.

Since the line we are looking for is perpendicular to this line, the slope of the new line will be the negative reciprocal of -3/2. The negative reciprocal is obtained by flipping the fraction and changing its sign:

slope of the new line = -2/3.

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

y - y1 = m(x - x1),

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

Substituting the values (1, -2) for (x1, y1) and -2/3 for m, the equation becomes:

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

Simplifying this equation gives:

y + 2 = -2/3x + 2/3.

To express the equation in the standard slope-intercept form (y = mx + b), we can isolate y:

y = -2/3x + 2/3 - 2.

After simplifying further, we obtain the equation of the line:

y = -2/3x - 4/3.