Create a colorful and appealing image depicting an abstract mathematical scene. Visualize a Cartesian plane with marked axes but no numeric indications. Show a line joining two distinct points, labeled as D and E. The line should be intersected at halfway by another line, symbolizing the perpendicular bisector. The bisector confidently intersects the y-axis at a point marked as (0,k). Keep the overall aesthetics soothing and scholarly without any text.

The perpendicular bisector of the line joining the points D(1,2) and E(5,4) meets the y-axis at the point (0,k).calculate k

First, find the midpoint of DE: ((1+5)/2,(2+4)/2) = (3,3)

Now, find the slope of the line DE: (4-2)/(5-1) = 2/4 = 1/2

You want a perpendicular line, so its slope will be -2

The equation of the desired line, using point-slope form:

(y-3)/(x-3) = -2
y-3 = -2x + 6
y = -2x + 9

The y-intercept is (0,9) so k=9

Well, to find the equation of the perpendicular bisector, we first need to find the midpoint of the line segment DE.

The midpoint is calculated by taking the average of the x-coordinates and the average of the y-coordinates of the given points. Let's do the math:

Midpoint (M) = ((1+5)/2, (2+4)/2)
= (6/2, 6/2)
= (3, 3)

Now that we have the midpoint, we can determine the slope of the line passing through points D and E. The slope is calculated by taking the difference in y-coordinates divided by the difference in x-coordinates:

Slope of DE = (4-2)/(5-1)
= 2/4
= 1/2

Since the perpendicular bisector is perpendicular to DE, its slope is the negative reciprocal of the slope of DE. The negative reciprocal of 1/2 is -2/1, which is -2.

Now, we have the slope (-2) and the point M(3, 3) on the perpendicular bisector. We can use the point-slope form to find the equation of the line:

y - y1 = m(x - x1)

Substituting in the values, we get:

y - 3 = -2(x - 3)

Expanding and rearranging:

y - 3 = -2x + 6

Simplifying:

y = -2x + 9

Now we need to find the y-intercept, which is the point where the line intersects the y-axis. We can do this by setting x = 0 and solving for y:

y = -2(0) + 9
y = 9

So, the y-intercept is (0, 9). Therefore, k = 9.

Thus, the value of k is 9.

To find the perpendicular bisector of a line, we need to find the midpoint of the line and then determine the slope of its perpendicular line.

Step 1: Finding the midpoint:
The midpoint of a line segment with endpoints (x₁, y₁) and (x₂, y₂) is given by the formulas:
Midpoint x-coordinate = (x₁ + x₂) / 2
Midpoint y-coordinate = (y₁ + y₂) / 2

Using these formulas, we can find the midpoint of line segment DE:
Midpoint x-coordinate = (1 + 5) / 2 = 6 / 2 = 3
Midpoint y-coordinate = (2 + 4) / 2 = 6 / 2 = 3

So, the midpoint of line segment DE is M(3, 3).

Step 2: Finding the slope of the perpendicular line:
The slope of the perpendicular line to a line with slope m is given by the negative reciprocal of m.

The slope of line DE can be found using the formula:
Slope (m) = (y₂ - y₁) / (x₂ - x₁)

Using the coordinates of points D and E:
Slope (m) = (4 - 2) / (5 - 1) = 2 / 4 = 1/2

The slope of the perpendicular bisector line will be the negative reciprocal of 1/2:
Perpendicular slope = -1 / (1/2) = -2/1 = -2

Step 3: Finding the equation of the line passing through the midpoint M with slope -2:
We can use the point-slope form of a line to find the equation y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope.

Using the point-slope form with point M(3, 3) and slope -2:
y - 3 = -2(x - 3)

Simplifying the equation:
y - 3 = -2x + 6
y = -2x + 9

Step 4: Finding the y-intercept point (0, k):
To find the y-intercept point, we substitute x = 0 into the equation of the line:
y = -2(0) + 9
y = 0 + 9
y = 9

So, the y-intercept point is (0, 9).

Therefore, k = 9.

To find the point at which the perpendicular bisector meets the y-axis, we need to identify the midpoint of the line segment DE first. The midpoint is the average of the x-coordinates and the average of the y-coordinates of the endpoints.

Coordinates of D: (1, 2)
Coordinates of E: (5, 4)

Midpoint formula:
Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)

Midpoint = ((1 + 5) / 2, (2 + 4) / 2)
Midpoint = (6 / 2, 6 / 2)
Midpoint = (3, 3)

So, the midpoint of DE is (3, 3).

Now, we can determine the slope of the line joining D and E using the slope formula:

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

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

The perpendicular bisector of a line has a slope that is the negative reciprocal of the original line's slope. Therefore, the perpendicular bisector has a slope of -2 (the negative reciprocal of 1/2).

To find the equation of the line perpendicular to DE and passing through the midpoint, we can use the point-slope form:

y - y1 = m(x - x1)

Using the midpoint (3, 3) and slope -2:

y - 3 = -2(x - 3)

Now, to find the value of k, we need to substitute x = 0 into the equation and solve for y:

y - 3 = -2(0 - 3)
y - 3 = -2(-3)
y - 3 = 6

Adding 3 to both sides:

y = 6 + 3
y = 9

Therefore, the point at which the perpendicular bisector meets the y-axis is (0, 9). So, k = 9.