The line pq has equation 3x – 2y = 12. p is a point with coordinates (6, 3) and q is the point (-2, k). 1 find the value of k. 2 find the equation of the line through p that is perpendicular to pq. the point r has coordinates (2, -5). 3 find the exact length pr.

Step-by-step explanation:

1. Put the point values in the equation where x and y are, then solve.

... 3(-2) -2k = 12

... -6 -2k = 12 . . . . eliminate parentheses

... -2k = 18 . . . . . . .add 6

... k = -9 . . . . . . . . . divide by -2

2. The perpendicular line will have a slope that is the negative reciprocal of the slope of the given line. The equation for it can be found by swapping the x- and y-coefficients and negating one of them. Then, to make the line go through point (h, k), replace the constant with zero and replace x with (x-h) and y with (y-k).

... 2(x -6) +3(y -3) = 0 . . . . . a line through P perpendicular to PQ

... 2x + 3y = 21

3. The Pythagorean theorem is used for finding the lengths of line segments. The difference in coordinates between P and R is ...

... P - R = (6, 3) - (2, -5) = (6-2, 3-(-5)) = (4, 8)

The length of segment PR can be considered to be the length of the hypotenuse of a right triangle with these leg lengths.

... ║PR║ = √(4² +8²) = √80

... ║PR║ = 4√5

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