Question

A line segment is drawn between (6,4) and (8,3). Find its gradient, midpoint and length.

Answer 1

To find the gradient of the line segment, we use the formula:

gradient = (change in y) / (change in x)

So, gradient = (3 - 4) / (8 - 6) = -1/2

To find the midpoint of the line segment, we use the formula:

midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)

So, midpoint = ((6 + 8) / 2, (4 + 3) / 2) = (7, 3.5)

To find the length of the line segment, we use the distance formula:

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

So, length = sqrt((8 - 6)^2 + (3 - 4)^2) = sqrt(5)

Therefore, the gradient of the line segment is -1/2, the midpoint is (7, 3.5), and the length is sqrt(5).

Answer 2

Gradient of the line segment:

The gradient of a line segment is given by the formula:

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

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

In this case, (x1, y1) = (6, 4) and (x2, y2) = (8, 3). Substituting into the formula, we get:

m = (3 - 4) / (8 - 6) = -1/2

Therefore, the gradient of the line segment is -1/2.

Midpoint of the line segment:

The midpoint of a line segment is given by the formula:

((x1 + x2) / 2, (y1 + y2) / 2)

In this case, (x1, y1) = (6, 4) and (x2, y2) = (8, 3). Substituting into the formula, we get:

((6 + 8) / 2, (4 + 3) / 2) = (7, 3.5)

Therefore, the midpoint of the line segment is (7, 3.5).

Length of the line segment:

The length of a line segment is given by the distance formula:

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

In this case, (x1, y1) = (6, 4) and (x2, y2) = (8, 3). Substituting into the formula, we get:

d = sqrt((8 - 6)^2 + (3 - 4)^2) = sqrt(2^2 + (-1)^2) = sqrt(5)

Therefore, the length of the line segment is sqrt(5).