Collinearity and Analytical Proofs | IEB Grade 11 Maths

Analytical proofs use coordinates to show that points lie on a line or that a shape has a certain property. The skill is choosing which gradients, lengths and midpoints to calculate, and writing a clear conclusion.

Collinear points

Points are collinear if they lie on the same straight line.

To prove $A$ , $B$ and $C$ are collinear, show $m_{A B} = m_{B C}$ . The shared point $B$ means the lines are the same line, not just parallel.

Example. $A (- 2; - 1)$ , $B (1; 1)$ , $C (7; k)$ are collinear:

$m_{A B} = \frac{2}{3} = m_{B C} = \frac{k - 1}{6} \Rightarrow k = 5$

Proving properties of triangles and quadrilaterals

Triangles

Right-angled: two sides have gradients with a product of $- 1$ .
Isosceles: two sides are equal in length.

Quadrilaterals (minimum conditions)

Parallelogram: diagonals have the same midpoint, or both pairs of opposite sides are parallel.
Rhombus: a parallelogram with two adjacent sides equal.
Rectangle: a parallelogram with a right angle.
Square: a rhombus with a right angle.
Kite: two pairs of adjacent sides equal.

Example. $A (- 2; 3)$ , $B (4; 5)$ , $C (6; - 1)$ , $D (0; - 3)$ . Both diagonals have midpoint $(2; 1)$ , so $A B C D$ is a parallelogram. $A B = B C = 40$ , so it is a rhombus. $m_{A B} \times m_{B C} = \frac{1}{3} \times (- 3) = - 1$ , so it is a square.

End each proof with a sentence that names the property you used.

Key terms

Collinear: Lying on the same straight line.
Kite: A quadrilateral with two pairs of equal adjacent sides.
Rhombus: A parallelogram with all four sides equal.

Common mistakes

Showing two gradients are equal without a shared point when proving collinearity.
Proving a parallelogram from only one pair of parallel sides.
Leaving out the conclusion sentence.
Calculating unnecessary lengths and running out of time.

Practice questions

Reveal steps one at a time, then tick the marks you earned.

Question 1

4 marks total

routine

$A (- 2; - 1)$ , $B (1; 1)$ and $C (7; k)$ are collinear. Determine $k$ .

Worked solution

Hidden - try the question first

Solution

$m_{A B} = \frac{1 + 1}{1 + 2} = \frac{2}{3}$
$m_{B C} = \frac{k - 1}{6}$
$\frac{k - 1}{6} = \frac{2}{3}$
$k = 5$

Question 2

5 marks total

routine

Prove that $P (1; 1)$ , $Q (4; 5)$ and $R (8; 2)$ form a right-angled isosceles triangle.

Worked solution

Hidden - try the question first

Solution

$m_{A B} = \frac{1 + 1}{1 + 2} = \frac{2}{3}$
$QR = 16 + 9 = 5$
$m_{P Q} = \frac{4}{3}$ and $m_{QR} = - \frac{3}{4}$
$m_{P Q} \times m_{QR} = - 1$ , so $\hat{Q} = 90°$
$P Q = QR$ and $\hat{Q} = 90°$ : right-angled isosceles

Question 3

6 marks total

complex

$A (- 3; 0)$ , $B (0; 4)$ , $C (3; 0)$ and $D (0; - 2)$ are the vertices of a quadrilateral.

(a) Prove that $A B C D$ is a kite. (4)

(b) Prove that the diagonals are perpendicular. (2)

Worked solution

Hidden - try the question first

Solution

(a) $A B = 9 + 16 = 5$ and $B C = 9 + 16 = 5$
$A D = 9 + 4 = 13$ and $C D = 9 + 4 = 13$
Two pairs of adjacent sides are equal, so $A B C D$ is a kite
(b) $A C$ lies on the $x$ -axis (horizontal) and $B D$ lies on the $y$ -axis (vertical)
So $A C ⊥ B D$

Question 4

9 marks total

problem-solving

$A (- 2; 3)$ , $B (4; 5)$ , $C (6; - 1)$ and $D (0; - 3)$ are the vertices of quadrilateral $A B C D$ .

(a) Prove that $A B C D$ is a parallelogram. (3)

(b) Prove that $A B C D$ is a rhombus. (2)

(d) Calculate the area of $A B C D$ . (2)

Worked solution

Hidden - try the question first

Solution

(a) $A B = 9 + 16 = 5$
$M_{B D} = (\frac{4 + 0}{2}; \frac{5 - 3}{2}) = (2; 1)$
The diagonals bisect each other, so $A B C D$ is a parallelogram
(b) $A B = 36 + 4 = 40$ and $B C = 4 + 36 = 40$
A parallelogram with adjacent sides equal is a rhombus
(c) $m_{A B} = \frac{1}{3}$ and $m_{B C} = - 3$ ; product $- 1$ , so $\hat{B} = 90°$
A rhombus with a right angle is a square
(d) Area $= (40)^{2}$
$= 40$ square units

FAQs

How do I prove that points are collinear?

Show that two gradients that share a point are equal.

What is the quickest proof of a parallelogram?

Show the diagonals have the same midpoint.

Do I need to write a conclusion?

Yes. The final mark is usually for stating what has been proved and why.

Collinearity and proving properties of shapes

Collinear points

Proving properties of triangles and quadrilaterals

Key terms

Common mistakes

Practice questions

FAQs

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