Cikgu Kai
Home
About Us
Articles
Study Resources
Support
Menu
Login
Sign Up
arrow_back
Back to Study Resources
Chapter 7: Coordinate Geometry
Form 4 Additional Mathematics Bab 7: Coordinate Geometry
7.1 Divisor of a Line Segment
A point P on line segment AB dividing it in the ratio m : n means AB is divided into (m + n) equal parts; AP : PB = m : n.
(p.176)
If m = n, P is the midpoint of AB.
(p.176)
Divisor formula: P(x, y) = ((nx
1
+ mx
2
)/(m + n), (ny
1
+ my
2
)/(m + n)) for A(x
1
, y
1
), B(x
2
, y
2
) with AP : PB = m : n.
(p.178)
Midpoint: M = ((x
1
+ x
2
)/2, (y
1
+ y
2
)/2).
(p.178)
Distance between two points: d = √((x
2
− x
1
)
2
+ (y
2
− y
1
)
2
).
(p.178)
To find an unknown ratio, equate one coordinate of the point to the divisor formula and solve for m/n.
(p.180)
7.2 Parallel Lines and Perpendicular Lines
Gradient m = (y
2
− y
1
)/(x
2
− x
1
) = tan θ, where θ is the angle with the positive x-axis (0° ≤ θ < 180°).
(p.185)
Two lines are parallel if and only if m
1
= m
2
.
(p.185)
Two lines are perpendicular if and only if m
1
m
2
= −1.
(p.185)
Gradient form: y = mx + c. Intercept form: x/a + y/b = 1 with gradient −b/a.
(p.186)
Equation of a line with gradient m through (x
1
, y
1
): y − y
1
= m(x − x
1
).
(p.188)
7.3 Areas of Polygons
Area of triangle ABC = ½|(x
1
y
2
+ x
2
y
3
+ x
3
y
1
) − (x
2
y
1
+ x
3
y
2
+ x
1
y
3
)| (shoelace formula, vertices in order).
(p.193)
The absolute value ensures the area is positive; if the vertices are clockwise the unsigned result is negative.
(p.193)
If the area of the triangle formed by three points is 0, the points are collinear.
(p.194)
Area of quadrilateral = ½|(x
1
y
2
+ x
2
y
3
+ x
3
y
4
+ x
4
y
1
) − (x
2
y
1
+ x
3
y
2
+ x
4
y
3
+ x
1
y
4
)|.
(p.195)
For an n-sided polygon with vertices in order, area = ½|(sum of x
i
y
i+1
) − (sum of x
i+1
y
i
)|.
(p.196)
7.4 Equations of Loci
A locus is the path of a moving point subject to given conditions.
(p.200)
Constant distance r from a fixed point A(x
1
, y
1
) (circle): (x − x
1
)
2
+ (y − y
1
)
2
= r
2
.
(p.200)
Distances from A and B in ratio m : n (PA/PB = m/n): ((x − x
1
)
2
+ (y − y
1
)
2
) / ((x − x
2
)
2
+ (y − y
2
)
2
) = m
2
/n
2
.
(p.201)
Equal distance from A and B (PA = PB): locus is the perpendicular bisector of AB; (x − x
1
)
2
+ (y − y
1
)
2
= (x − x
2
)
2
+ (y − y
2
)
2
.
(p.202)
Sponsored