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Geometry (HS pathway)

Coordinate Geometry: Proofs and Partitioning Segments

Geometry

  • ✓By the end of this lesson students will be able to apply the distance formula to calculate lengths of segments in the coordinate plane.
  • ✓By the end of this lesson students will be able to use the slope formula to determine if lines are parallel or perpendicular.
  • ✓By the end of this lesson students will be able to perform coordinate proofs to verify geometric properties of figures (e.g., parallelograms, right triangles).
  • ✓By the end of this lesson students will be able to find the coordinates of a point that partitions a directed line segment in a given ratio.

Key concepts

Distance Formula

The distance formula is used to find the length of a line segment connecting two points (x1, y1) and (x2, y2) in the coordinate plane. It is derived directly from the Pythagorean theorem.

d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Slope Formula

The slope of a line measures its steepness and direction. It is defined as the ratio of the 'rise' (change in y-coordinates) to the 'run' (change in x-coordinates) between any two distinct points on the line.

m = (y2 - y1) / (x2 - x1)
Parallel Lines

In the coordinate plane, two non-vertical lines are parallel if and only if they have the same slope. All vertical lines are parallel to each other.

m1 = m2
Perpendicular Lines

In the coordinate plane, two non-vertical lines are perpendicular if and only if the product of their slopes is -1. This means their slopes are negative reciprocals of each other. A horizontal line and a vertical line are also perpendicular.

m1 * m2 = -1 (or m1 = -1/m2)
Midpoint Formula

The midpoint formula finds the coordinates of the point that is exactly halfway between two given points (x1, y1) and (x2, y2). It is the average of the x-coordinates and the average of the y-coordinates.

M = ((x1 + x2)/2, (y1 + y2)/2)
Partitioning a Segment in a Given Ratio

To find the coordinates of a point P that partitions a directed line segment from A(x1, y1) to B(x2, y2) in a given ratio a:b, you can think of it as moving a fraction of the way along the segment. The fraction of the total length is a / (a+b).

P(x, y) where x = x1 + (a / (a+b)) * (x2 - x1) and y = y1 + (a / (a+b)) * (y2 - y1)

Key facts to remember

  • 1The Distance Formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
  • 2The Slope Formula: m = (y2 - y1) / (x2 - x1)
  • 3Parallel lines have equal slopes (m1 = m2).
  • 4Perpendicular lines have slopes that are negative reciprocals (m1 * m2 = -1).
  • 5The Midpoint Formula: M = ((x1 + x2)/2, (y1 + y2)/2)
  • 6To partition a segment from (x1, y1) to (x2, y2) in ratio a:b, use P(x, y) where x = x1 + (a / (a+b)) * (x2 - x1) and y = y1 + (a / (a+b)) * (y2 - y1).
  • 7Coordinate proofs use algebraic methods (distance, slope, midpoint formulas) to prove geometric properties.

Worked examples

Example 1

Prove that the quadrilateral with vertices A(-3, 2), B(2, 5), C(6, 0), and D(1, -3) is a parallelogram.

ITo prove a quadrilateral is a parallelogram, we can show that both pairs of opposite sides are parallel. We will use the slope formula.
IISlope of AB (m_AB): (5 - 2) / (2 - (-3)) = 3 / 5
IIISlope of BC (m_BC): (0 - 5) / (6 - 2) = -5 / 4
IVSlope of CD (m_CD): (-3 - 0) / (1 - 6) = -3 / -5 = 3 / 5
VSlope of DA (m_DA): (2 - (-3)) / (-3 - 1) = 5 / -4 = -5 / 4
VICompare slopes: m_AB = 3/5 and m_CD = 3/5. Since m_AB = m_CD, segment AB is parallel to segment CD.
VIICompare slopes: m_BC = -5/4 and m_DA = -5/4. Since m_BC = m_DA, segment BC is parallel to segment DA.

Answer

Since both pairs of opposite sides are parallel (AB || CD and BC || DA), the quadrilateral ABCD is a parallelogram.

Alternatively, you could use the distance formula to show that opposite sides are congruent, or the midpoint formula to show that the diagonals bisect each other.

Example 2

Determine if the triangle with vertices P(1, 4), Q(3, -2), and R(-5, -6) is a right triangle.

ITo determine if a triangle is a right triangle, we can check if any two sides are perpendicular. This means the product of their slopes should be -1.
IISlope of PQ (m_PQ): (-2 - 4) / (3 - 1) = -6 / 2 = -3
IIISlope of QR (m_QR): (-6 - (-2)) / (-5 - 3) = (-6 + 2) / (-8) = -4 / -8 = 1/2
IVSlope of RP (m_RP): (4 - (-6)) / (1 - (-5)) = (4 + 6) / (1 + 5) = 10 / 6 = 5/3
VCheck for perpendicular slopes:
VIm_PQ * m_QR = (-3) * (1/2) = -3/2 (Not -1)
VIIm_QR * m_RP = (1/2) * (5/3) = 5/6 (Not -1)
VIIIm_RP * m_PQ = (5/3) * (-3) = -5 (Not -1)

Answer

Since no two sides have slopes that are negative reciprocals (or whose product is -1), the triangle PQR is not a right triangle.

Another method would be to use the distance formula to find the lengths of all three sides and then check if a^2 + b^2 = c^2 (Pythagorean theorem).

Example 3

Find the coordinates of point P that partitions the directed line segment from A(-4, 6) to B(10, -8) in the ratio 3:4.

IIdentify the coordinates of the starting point A(x1, y1) = (-4, 6) and the ending point B(x2, y2) = (10, -8).
IIIdentify the ratio a:b = 3:4, so a = 3 and b = 4.
IIICalculate the fraction of the total length: a / (a+b) = 3 / (3+4) = 3/7.
IVUse the partitioning formula for the x-coordinate:
Vx_P = x1 + (a / (a+b)) * (x2 - x1)
VIx_P = -4 + (3/7) * (10 - (-4))
VIIx_P = -4 + (3/7) * (14)
VIIIx_P = -4 + 6
9x_P = 2
10Use the partitioning formula for the y-coordinate:
11y_P = y1 + (a / (a+b)) * (y2 - y1)
12y_P = 6 + (3/7) * (-8 - 6)
13y_P = 6 + (3/7) * (-14)
14y_P = 6 - 6
15y_P = 0

Answer

The coordinates of point P are (2, 0).

A directed line segment means the order of the points matters. Starting from A and going towards B is different from starting from B and going towards A.

Common mistakes

  • ✗Confusing the distance formula with the slope formula, or mixing up the operations (e.g., adding in distance formula instead of subtracting).
  • ✗Incorrectly calculating the change in x or y for slope, especially with negative numbers (e.g., (y1 - y2) / (x2 - x1)).
  • ✗Forgetting to take the negative reciprocal when checking for perpendicular slopes.
  • ✗For partitioning segments, incorrectly setting up the ratio fraction or forgetting to add the initial coordinate (x1 or y1) to the calculated change.
  • ✗Algebraic errors when squaring numbers, handling negative signs, or simplifying fractions.

Exam tips

  • ★Always draw a diagram for coordinate geometry problems. Visualizing the points and segments can help prevent errors and confirm your calculations.
  • ★Clearly label your points (x1, y1), (x2, y2) before applying formulas to avoid substitution mistakes.
  • ★Show all your steps in coordinate proofs. This allows for partial credit and helps you track your work.
  • ★Know the properties of common geometric figures (e.g., parallelogram, rectangle, right triangle) to choose the most efficient method for your proof.
  • ★Double-check your calculations, especially when dealing with negative numbers and fractions.

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