S3 Coordinate Geometry

Question 1

Slope of a line given two points (a, b) (c, d)

Answer 1

d-b/c-a

Question 2

Distance of two points (a, b) (c, d)

Answer 2

[(c-a)²+(d-b)²]^1/2

Question 3

Slope of a line given inclination a

Answer 3

tan(a)

Question 4

What is the relation between the slopes of two parallel lines?

Answer 4

They are equal.

Question 5

What is the relation between the slopes of two perpendicular lines?

Answer 5

They multiply to -1/ one is the negative reciprocal of the other.

Question 6

Centroid of triangle given vertices (a, b) (c, d) (e, f)

Answer 6

(a+c+e/3, b+d+f/3)

Question 7

Mid-point of segment with end-points (a, b) (c, d)

Answer 7

(a+c/2, b+d/2)

Question 8

x-intercept of line passing through (0, a) (b, 0):
a. a
b. b
c. (0, a)
d. (b, 0)

Answer 8

a

Question 9

y-intercept of line passing through (0, a) (b, 0):
a. a
b. b
c. (0, a)
d. (b, 0)

Answer 9

b

Question 10

Define a line segment with end-points A (a, b) and B (c, d). Define M on the segment such that AM:BM = p:q. Find the coordinates of M.

Answer 10

(pc+qa/p+q, pd+qb/p+q)

Question 11

Find the coordinates of the circumcentre of a triangle with vertices O (0, 0), A (0, a), B (b, 0).

Answer 11

(b/2, a/2)
OA is vertical. The perpendicular bisector of OA is horizontal and passes through the midpoint of OA (0, a/2). Therefore, the y-coordinate of the ciucumcentre is a/2.
Similarly, OB is horizontal. Its perpendicular bisector is vertical passing through (b/2, 0). Since the circumcentre must lie on both perpendicular bisectors, its coordinates are (b/2, a/2).

Question 12

Find the coordinates of the orthocentre of a triangle with vertices O (0, 0), A (a, a), B (b, 0).

Answer 12

OB is horizontal: the slope of the height with base as OB is vertical. The orthocentre has x-coordinate of a. Let y be the y-coordinate of the orthocentre.
slope of OA=1: the slope of the height with base as OA is -1.
requirement: (a, y) and (b, 0) have a slope of -1.
y = b-a
coordinates of orthocentre = (a, b-a)