Analytical Geometry Study Notes

Distance between Two Points

Explanation:

The distance between two points on the Cartesian plane measures the straight-line length connecting them. This formula comes from the Pythagorean Theorem.

Important Facts:

• You must know the coordinates of both points.
• The order of subtraction does not affect the result because the terms are squared.

Formula:

d = sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Example:

Calculate the distance between A(2 ; 3) and B(7 ; 11).

Step 1: Identify the coordinates.

A(2,3),quad B(7,11)

Step 2: Write the formula.

d = sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Step 3: Substitute the values.

d = sqrt{(7-2)^2 + (11-3)^2}

Step 4: Simplify.

= sqrt{5^2 + 8^2}

= sqrt{25 + 64}

= sqrt{89}

Step 5: Final answer.

d = sqrt{89}

Remember:

• Always write final answers in simplest surd form unless asked for a decimal.

Common Mistake:

• Swapping coordinates but not squaring may lead to negative or incorrect results.

Gradient of a Line

Explanation:

The gradient of a line describes its steepness and direction. It is calculated as the ratio of the vertical change to the horizontal change between two points.

Important Facts:

• The gradient is denoted by m.
• Parallel lines have equal gradients.
• A positive gradient means the line rises as it moves from left to right; a negative gradient means it falls.

Formula:

m = frac{y_2 - y_1}{x_2 - x_1}

Example:

Find the gradient of the line passing through Q(–4 ; –6) and R(12 ; 2).

Step 1: Identify the coordinates.

Q(-4,-6),quad R(12,2)

Step 2: Write the formula.

m = frac{y_2 - y_1}{x_2 - x_1}

Step 3: Substitute the values.

m = frac{2 - (-6)}{12 - (-4)}

Step 4: Simplify.

= frac{2 + 6}{12 + 4}

= frac{8}{16}

= frac{1}{2}

Step 5: Final answer.

m = frac{1}{2}

Exam Tip:

• Always check for division by zero (vertical lines have undefined gradients).

Equation of a Straight Line

Explanation:

The equation of a straight line relates x and y so that all (x, y) pairs on the line satisfy the equation.

Important Facts:

• The standard form of the equation is y = mx + c, where m is the gradient and c is the y-intercept.
• The y-intercept is found where x=0.

Formula:

y = mx + c

Example:

Find the equation of the line that passes through Q(–4 ; –6) with m = ½.

Step 1: Identify given values.

Q(-4, -6),quad m = frac{1}{2}

Step 2: Write the formula.

y = mx + c

Step 3: Substitute the gradient and the point.

-6 = frac{1}{2}(-4) + c

Step 4: Simplify.

-6 = -2 + c

c = -6 + 2

c = -4

Step 5: Write the final equation.

y = frac{1}{2}x - 4

Common Mistake:

• Mixing up x- and y-coordinates when substituting into the equation.

Angle of Inclination

Explanation:

The angle of inclination ( θ ) of a line is the angle measured counterclockwise from the positive x-axis to the line.

Important Facts:

• 0 degrees means the line is parallel to the x-axis (horizontal).
• Angles between 0 and 180 degrees.
• The angle of inclination is linked to the gradient: m = tan θ.

Formula:

tan θ = m

Example:

Find the angle of inclination of a line with m = ½.

Step 1: Write the formula.

tan θ = m

Step 2: Substitute the value for m.

tan θ = frac{1}{2}

Step 3: Calculate θ.

θ = arctan(frac{1}{2})

θ ≈ 26,57°

Step 4: State the final answer.

Angle of inclination = 26,57°

Exam Tip:

• Use your calculator in degree mode for all CAPS examination calculations.

Angle between Two Lines

Explanation:

The angle between two straight lines measures how far one line turns to coincide with the other.

Important Facts:

• The formula involves the gradients of both lines.
• If both gradients are known, the acute angle θ between them is found by:

Formula:

tan θ = frac{|m_1 - m_2|}{1 + m_1 m_2}

Example:

Find the angle between two lines with m₁ = 2 and m₂ = –1.

Step 1: Write the formula.

tan θ = frac{|m_1 - m_2|}{1 + m_1 m_2}

Step 2: Substitute the values.

tan θ = frac{|2 - (-1)|}{1 + (2)(-1)}

Step 3: Simplify.

= frac{|2 + 1|}{1 - 2}

= frac{3}{-1}

= -3

Step 4: Calculate θ.

θ = arctan(3)

θ ≈ 71,57°

Parallel and Perpendicular Lines

Explanation:

Parallel lines have the same gradient, while perpendicular lines have gradients that multiply to –1.

Important Facts:

• Parallel: m₁ = m₂
• Perpendicular: m₁ × m₂ = –1

Example:

If line 1 has gradient m₁ = 3, what is the gradient of a line perpendicular to it?

Step 1: Write the relationship for perpendicular gradients.

m_1 × m_2 = –1

Step 2: Substitute the value for m₁.

3 × m_2 = –1

Step 3: Solve for m₂.

m_2 = –frac{1}{3}

Common Mistake:

• Forgetting to take the negative reciprocal for perpendicular lines.

Application:

• Used to show lines in geometric figures are parallel or perpendicular, such as in rectangles or proving rhombus properties.

Midpoint Theorem

Explanation:

The midpoint of a line segment divides it into two equal parts. The coordinates of the midpoint are the averages of the corresponding coordinates of the endpoints.

Formula:

For points P( x_1 ; y_1 ) and Q( x_2 ; y_2 ),

M(x_m; y_m) = left( frac{x_1 + x_2}{2}; frac{y_1 + y_2}{2} right)

Example:

Find the midpoint between P(–1 ; 8) and Q(–4 ; –6).

Step 1: Identify the coordinates.

P(-1, 8),quad Q(-4, -6)

Step 2: Write the formula.

M = left( frac{-1 + (-4)}{2}, frac{8 + (-6)}{2} right)

Step 3: Simplify.

= left( frac{-5}{2}, frac{2}{2} right)

= left( -2,5;\ 1 right)

Step 4: State the answer.

Midpoint = (-2,5 ; 1)

Exam Tip:

• Double check arithmetic when adding or dividing negative numbers.

Coordinate Geometry Proofs

Explanation:

Coordinate geometry proofs use algebraic methods with coordinates, gradients, distances and midpoints to prove geometric properties of figures on the Cartesian plane.

Common Proof Types:

• Prove a quadrilateral is a parallelogram by showing both pairs of opposite sides are parallel or equal.
• Prove a triangle is isosceles by showing two sides are equal using the distance formula.
• Prove a point is the midpoint of a segment using the midpoint formula.

Example:

Show that triangle PQR with P(–1 ; 8), Q(–4 ; –6), and R(12 ; 2) is isosceles.

Step 1: Find PQ.

PQ = sqrt{(–4 – (–1))^2 + (–6 – 8)^2}

= sqrt{(–3)^2 + (–14)^2}

= sqrt{9 + 196}

= sqrt{205}

Step 2: Find PR.

PR = sqrt{(12 – (–1))^2 + (2 – 8)^2}

= sqrt{13^2 + (–6)^2}

= sqrt{169 + 36}

= sqrt{205}

Step 3: Statement.

Since PQ = PR, triangle PQR is isosceles.

Exam Tip:

• State clearly what you are proving and reference your calculation.
• Use correct mathematical notation for all steps.

Conclusion

Analytical Geometry in Grade 12 covers essential tools for exploring and proving properties of geometric figures using algebraic methods. Mastery of these techniques is important for examinations. Always show your working clearly, use step by step reasoning, and double-check calculations for accuracy.