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To find the perpendicular distance you need to find the equation of the line the two points (1, 3.25) and (6, -0.5), and the equation of the line through the point (7, 5) that is perpendicular to this line.

With these equations, you can find the common point (using simultaneous equations) where they intercept and thus the distance (using Pythagoras) between this point and the point (7, 5).

To do this you need:

The gradient m of a line between two points (x0, y0) and (x1, y1) is given by:

[1] m = change_in_y/change_in_x = (y1 - y0)/(x1 - x0)

The equation of a line with gradient m thorough a point (x0, y0) is given by:

[2] y - y0 = m(x - x0).

The gradient m' of a perpendicular line to a line with gradient m is such that:

[3] mm' = -1, ie m' = -1/m

The distance between two points (x0, y0) and (x1, y1) is given using Pythagoras by:

[4] distance = √((x1 - x0)^2 + (y1-y0)^2)

Using the above the problem can be solved:

1) Find the equation of the line between the two points (1, 3.25) and (6, -0.5), using [1] and [2] above;

2) Find the equation of the perpendicular line going through the point (7, 5), using [3] and [2] above;

3) Solve the simultaneous equations formed from (1) and (2) to get the point of intercept

4) Find the distance from this point to (7, 5) using [4] above.

Have a go at a solution using the above before reading any further.

This is the solution using the above method:

1) gradient m = ((-0.5) - (3.25))/(6 - 1) = -3.75/5 = -0.75 = -3/4

→ line is (using the fraction):

y - (-0.5) = -(3/4)(x - 6)

→ 4y + 2 = -3x + 18

→ 4y + 3x = 16

2) gradient m' = -1/(-0.75) = -1/(-3/4) = 4/3 = 1.333...

→ line is (using the fraction):

y - 5 = (4/3)(x - 7)

→ 3y - 15 = 4x - 28

→ 3y - 4x = -13

3) solving to find the common point (point of intercept):

(1)x3 → 12y + 9x = 48

(2)x4 → 12y - 16x = -52

subtracting second from first:

→ 25x = 100

→ x = 4

Substituting back into (1) gives:

4y + 3x4 = 16

→ 4y +12 = 16

→ 4y = 4

→ y = 1

Checking by substituting in (2) gives:

3y - 4x = -13

3x1 - 4x4 = 3 - 16 = -13 as required

→ point of intercept is (4, 1)

4) Distance required is from (4,1) to (7, 5)

→ distance = √((7 - 4)^2 + (5 - 1)^2) = √(3^2 + 4^2) = √(9 + 16) = √25 = 5

Q: What is the perpendicular distance from the point of 7 5 that meets the line of 1 3.25 and 6 -0.5 at right angles on the Cartesian plane showing all work leading to the answer?

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