Coordinate Geometry

Grade 10 · Math · Study Guide

Explore lines, distances, midpoints, and circles using the coordinate plane.

Coordinate geometry (analytic geometry) bridges algebra and geometry by placing geometric figures on the Cartesian plane. Every geometric relationship can be expressed as an equation or formula.

The distance between two points (x₁, y₁) and (x₂, y₂) is found using the Pythagorean theorem: d = √[(x₂−x₁)² + (y₂−y₁)²]. The midpoint of the segment joining them is M = ((x₁+x₂)/2, (y₁+y₂)/2).

The slope of a line through two points is m = (y₂−y₁)/(x₂−x₁). Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals (m₁ × m₂ = −1). The equation of a line can be written in slope-intercept form y = mx + b, or in the more general standard form ax + by + c = 0.

The equation of a circle with centre (h, k) and radius r is (x − h)² + (y − k)² = r². Expanding gives the general form x² + y² + Dx + Ey + F = 0, which can be converted back to standard form by completing the square.

The distance from a point (x₀, y₀) to a line ax + by + c = 0 is |ax₀ + by₀ + c| / √(a² + b²). This formula is essential for finding the radius of a circle tangent to a given line.

Key terms

Slope
The ratio of vertical change to horizontal change between two points on a line: m = Δy/Δx.
Midpoint Formula
The midpoint of segment joining (x₁,y₁) and (x₂,y₂) is ((x₁+x₂)/2, (y₁+y₂)/2).
Distance Formula
The distance between (x₁,y₁) and (x₂,y₂) is √[(x₂−x₁)²+(y₂−y₁)²].
Standard Form of a Circle
(x−h)²+(y−k)²=r², where (h,k) is the centre and r is the radius.
Collinear Points
Three or more points that lie on the same straight line.
Perpendicular Bisector
A line perpendicular to a segment passing through its midpoint; all points on it are equidistant from both endpoints.

Flashcards

Distance Formula
d = √[(x₂−x₁)²+(y₂−y₁)²]
Midpoint Formula
M = ((x₁+x₂)/2, (y₁+y₂)/2)
Slope Formula
m = (y₂−y₁)/(x₂−x₁)
Parallel Slopes
Parallel lines have equal slopes: m₁ = m₂.
Perpendicular Slopes
Perpendicular lines satisfy m₁ × m₂ = −1.
Circle Standard Form
(x−h)²+(y−k)²=r², centre (h,k), radius r.
Point-to-Line Distance
Distance from (x₀,y₀) to ax+by+c=0 is |ax₀+by₀+c|/√(a²+b²).
Completing the Square (circle)
Rewrite x²+Dx as (x+D/2)²−(D/2)² to find the centre and radius from the general form.

Questions

  1. What is the distance between points A(1, 2) and B(5, 6)?

    1. 4√2
    2. 8
    3. 6
    4. 4
    Show answer

    Answer: 4√2

    d = √[(5−1)²+(6−2)²] = √[16+16] = √32 = 4√2.

  2. Two lines are perpendicular if the product of their slopes equals ______.

    Show answer

    Answer: -1

    The product of slopes of perpendicular lines is always −1; this is a fundamental property.

  3. What is the centre of the circle x² + y² − 6x + 4y − 3 = 0?

    1. (3, −2)
    2. (−3, 2)
    3. (6, −4)
    4. (−6, 4)
    Show answer

    Answer: (3, −2)

    Complete the square: (x²−6x+9)+(y²+4y+4)=16 → (x−3)²+(y+2)²=16, so centre is (3,−2).

  4. The midpoint of a segment is always equidistant from both endpoints.

    Show answer

    Answer: True

    By definition, the midpoint divides the segment into two equal halves, so it is equidistant from both endpoints.

  5. A line has equation 3x − 4y + 12 = 0. What is the distance from the origin to this line?

    1. 12/5
    2. 12
    3. 3
    4. 4
    Show answer

    Answer: 12/5

    Distance = |3(0)−4(0)+12|/√(9+16) = 12/5.

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