Grade 11 Maths · Analytical Geometry

Calculate the inclination of a line

Find the angle a line makes with the x-axis using tan θ = m, for example a gradient of 1 gives 45°.

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Grade 11 Analytical geometry: angle of inclinationKevinmathscience

Grade 11 Analytical geometry: angle of inclination PracticeKevinmathscience

Grade 11 Analytical Geometry Angle of Inclination Introduction and practiceMiss Martins Maths and Science

Angle of Inclination: Find the angle between two lines | 11 Analytical GeometryMiss Martins Maths and Science

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Question 4.7 · May/June 2025 · Paper 2 · National (DBE) · 5 marks

In the diagram, M is the centre of the circle having equation

(x - 3)^{2} + y^{2} = 25

E (- 1; 3)

and T are points on the circle.

EC is a tangent to the circle at E and cuts the x-axis at D.

E D = \frac{15}{4}

units. MT is produced to meet the tangent at

C (- 7; p)

4.7 If ET is drawn, calculate the size of

E \hat{T} M

.(5)

Work it out on paper first. Then mark yourself with the official memo, one tick at a time.

More past paper questions: calculate the inclination of a line

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3. Question 3.3.1.1 · March 2016 · Provincial · 2 marks
3.3 In the diagram below, $A (- 1; 2)$ , $B (x; y)$ and $C (1; - 1)$ are the vertices of a triangle. D is the x-intercept and the midpoint of AB. $B \hat{A} C$ is $78, 6 9^{\circ}$ .
3.3.1.1 Calculate the size of angle $θ$ .(2)
Show the memo (2 ticks)Memo
1. ✓
  Reference angle 56,31° $m_{A C} = \frac{- 1 - 2}{1 - ( - 1 )} = - \frac{3}{2}$
  Reference angle $= 56, 3 1^{\circ}$
  1 mark · Calculate the inclination of a line
2. ✓
  θ = 45° $α = 18 0^{\circ} - 56, 3 1^{\circ} = 123, 6 9^{\circ}$
  $θ = 123, 6 9^{\circ} - 78, 6 9^{\circ} = 4 5^{\circ}$
  1 mark · Calculate the inclination of a line
4. Question 4.7 · March 2015 · Provincial · 5 marks
$P (- 6; 1)$ , $Q (6; - 5)$ and $R (4; 5)$ are the co-ordinates of $Δ PQR$ . C is the midpoint of QR. A, B and C are the intercepts of lines PQ and QR respectively.
$B \hat{Q} C = α$ , $P \hat{A} X = θ_{1}$ and $R \hat{C} X = θ_{2}$ .
4.7 Calculate the size of $α$ .(5)
Show the memo (4 ticks)Memo
1. ✓
  Reference angle 26,6° $tan θ_{1} = m_{PQ} = - \frac{1}{2}$ , reference angle $= 26, 6^{\circ}$
  1 mark · Calculate the inclination of a line
2. ✓
  Reference angle 78,7° $m_{QR} = \frac{5 - ( - 5 )}{4 - 6} = - 5$
  $tan θ_{2} = - 5$ , reference angle $= 78, 7^{\circ}$
  1 mark · Calculate the inclination of a line
3. ✓
  θ2 = 101,3° $θ_{2} = 18 0^{\circ} - 78, 7^{\circ} = 101, 3^{\circ}$
  1 mark · Calculate the inclination of a line
4. ✓
  α = 52,1° $θ_{1} = 18 0^{\circ} - 26, 6^{\circ} = 153, 4^{\circ}$
  $α = θ_{1} - θ_{2} = 153, 4^{\circ} - 101, 3^{\circ} = 52, 1^{\circ}$
  2 marks · Calculate the inclination of a line
5. Question 4.2.1 · March 2018 · Provincial · 3 marks
4.2 In the diagram $P (3; 7), Q (- 3; 4)$ and $R (1; - 6)$ are vertices of a triangle.
4.2.1 Calculate the angle of inclination of the line PS.(3)
Show the memo (3 ticks)Memo
1. ✓
  Gradient of PS (using PQ) $m_{PQ} = \frac{7 - 4}{3 - ( - 3 )} = \frac{3}{6} = \frac{1}{2}$
  1 mark · Calculate gradient between two points
2. ✓
  Tan theta $tan θ = \frac{1}{2}$
  1 mark · Calculate the inclination of a line
3. ✓
  Calculate angle $θ = 26, 5 7^{\circ}$
  1 mark · Calculate the inclination of a line
6. Question 2.2 · June 2024 · Paper 2 · Provincial · 3 marks
In the diagram below, $A (- 2; 4)$ , $B (- 2; - 1)$ and $C (p; - 4)$ are the vertices of $Δ A BC$ . $Q$ is the x-intercept of the line $A C$ .
$θ$ is the angle of inclination of $A C$ and $B \hat{A} C = 3 7^{\circ}$ .
2.2 Calculate the equation of line $A C$ .(3)
Show the memo (3 ticks)Memo
1. ✓
  Gradient from the inclination $m = tan 12 7^{\circ} = - 1, 33$
  1 mark · Calculate the inclination of a line
2. ✓
  Substitution of A $y - 4 = - 1, 33 (x + 2)$
  1 mark · Find the equation of a line
3. ✓
  Equation of AC $y = - 1, 33 x - 2, 66 + 4$ $y = - 1, 33 x + 1, 34$
  1 mark · Find the equation of a line

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