Theme 9: Analytical geometry
G10 – Mathematics – Study Guide 2/2 OR:
Worked example 7
We can also prove that AB + BC = AC (distance method): _______________
AB = √ ( 5 – 2) 2 + ( 2 – (–1)) 2
The points A(-6; 2), B(2; -2) and C(-3; p) are collinear. Find the value of p.
_
=√ 9 + 9
Solution
_
=√ 18
If A, B and C lie on the same straight line, the gradient of AB is the same as the gradient of BC.
_
= 3√ 2
_______________
∴
BC = √ (7 – 5) 2 + (4 – 2) 2
𝑝– (–2) –2 – 2 ∴ ________ = ________ –3 – 2 2 – (–6)
_
=√ 4 + 4 _
= 2√ 2
_______________
_
=√ 25 + 25 _
=√ 50 _
_
_
𝑝 + 2
∴
–4 ______ ___ = –5 8
∴
1 𝑝 = __ 2
∴ 8𝑝 + 16 = 20
AC = √ (7 – 2) 2 + ( 4 –(–1)) 2 = 5√ 2
𝑚AB = 𝑚BC
_
∴ AB + BC = 3 √ 2 + 2 √ 2 = 5 √ 2 = AC Therefore A, B and C are collinear.
1.
Determine whether the following points lie on the same straight line: A(–1; 2), B(2; 5) and C(0; 3). Use the gradient method.
3.
Determine whether the following points lie on the same straight line: A(–1; –1), B(2; 8), C(3; 11) and D(1; 5).
2.
20
Exercise 9.4: Points on a straight line (collinear points)
Determine whether the following points lie on the same straight line: A(1; 5), B(2; 7) and C(3; 6). Use the distance method.