Grade: Grade 10 Subject: Mathematics Unit: Coordinate Geometry SAT: Geometry+Trigonometry ACT: Math

Coordinate Proofs

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What is a Coordinate Proof?

A coordinate proof uses the coordinate plane and algebraic methods to prove geometric theorems. By placing figures on a coordinate system, we can use formulas (distance, midpoint, slope) to verify properties like equal lengths, parallel lines, perpendicularity, and more.

Strategic Placement of Figures

The key to an effective coordinate proof is placing the figure conveniently on the coordinate plane:

Placement Strategies:
  • Use the origin: Place one vertex at (0, 0) to simplify calculations
  • Align with axes: Place one side along the x-axis or y-axis
  • Use symmetry: Center symmetric figures at the origin
  • Use variables: Let side lengths be variables (a, b, c) for general proofs

Essential Tools for Coordinate Proofs

To Prove... Use... Condition
Segments are equal Distance formula Distances are equal
Lines are parallel Slope formula Slopes are equal
Lines are perpendicular Slope formula Slopes are negative reciprocals (m₁ · m₂ = -1)
Point is on a line Substitution Point satisfies line equation
Segments bisect each other Midpoint formula Midpoints are the same
Points are collinear Slope formula Same slope between all pairs

Slope Formula Review

The slope m of a line through points (x₁, y₁) and (x₂, y₂) is:

m = (y₂ - y₁) / (x₂ - x₁)

Common Coordinate Proof Setups

Figure Suggested Placement Vertices
Right Triangle Right angle at origin, legs on axes (0,0), (a,0), (0,b)
Isosceles Triangle Base on x-axis, centered at origin (-a,0), (a,0), (0,h)
Rectangle Corner at origin, sides on axes (0,0), (a,0), (a,b), (0,b)
Square Corner at origin, sides on axes (0,0), (a,0), (a,a), (0,a)
Parallelogram One vertex at origin (0,0), (a,0), (a+b,c), (b,c)
SAT/ACT Connection: Coordinate proofs combine algebra and geometry skills. On the SAT, you may need to verify properties of shapes given vertex coordinates, find missing vertices, or determine relationships between lines. Understanding how to use slope and distance together is essential.

💡 Examples

Example 1: Proving a Quadrilateral is a Parallelogram

Problem: Prove that ABCD with A(0,0), B(4,0), C(6,3), D(2,3) is a parallelogram.

Solution:

A parallelogram has opposite sides parallel. We'll show AB || DC and AD || BC using slopes.

Step 1: Find slope of AB

m_AB = (0-0)/(4-0) = 0

Step 2: Find slope of DC

m_DC = (3-3)/(6-2) = 0

Since m_AB = m_DC = 0, AB || DC

Step 3: Find slope of AD

m_AD = (3-0)/(2-0) = 3/2

Step 4: Find slope of BC

m_BC = (3-0)/(6-4) = 3/2

Since m_AD = m_BC = 3/2, AD || BC

Conclusion: ABCD is a parallelogram because both pairs of opposite sides are parallel.

Example 2: Proving Diagonals are Perpendicular

Problem: Prove that the diagonals of rhombus PQRS with P(0,2), Q(3,4), R(6,2), S(3,0) are perpendicular.

Solution:

The diagonals are PR and QS. Lines are perpendicular if their slopes are negative reciprocals.

Step 1: Find slope of PR

m_PR = (2-2)/(6-0) = 0/6 = 0 (horizontal line)

Step 2: Find slope of QS

m_QS = (0-4)/(3-3) = -4/0 = undefined (vertical line)

Step 3: Analyze

A horizontal line (slope 0) is always perpendicular to a vertical line (undefined slope).

Conclusion: The diagonals PR and QS are perpendicular.

Example 3: Proving a Triangle is a Right Triangle

Problem: Prove that triangle with vertices A(1,1), B(4,1), C(1,5) is a right triangle.

Solution:

A right triangle has two perpendicular sides.

Step 1: Find slope of AB

m_AB = (1-1)/(4-1) = 0/3 = 0

Step 2: Find slope of AC

m_AC = (5-1)/(1-1) = 4/0 = undefined

Step 3: Check perpendicularity

AB is horizontal (slope 0) and AC is vertical (undefined slope), so AB ⊥ AC.

Conclusion: Triangle ABC is a right triangle with the right angle at vertex A.

Example 4: Proving the Midpoint Theorem

Problem: In triangle with vertices A(0,0), B(6,0), C(4,8), prove that the segment connecting the midpoints of AB and AC is parallel to BC and half its length.

Solution:

Step 1: Find midpoint M of AB

M = ((0+6)/2, (0+0)/2) = (3, 0)

Step 2: Find midpoint N of AC

N = ((0+4)/2, (0+8)/2) = (2, 4)

Step 3: Find slope of MN

m_MN = (4-0)/(2-3) = 4/(-1) = -4

Step 4: Find slope of BC

m_BC = (8-0)/(4-6) = 8/(-2) = -4

Since m_MN = m_BC = -4, MN || BC

Step 5: Compare lengths

MN = √[(2-3)² + (4-0)²] = √[1 + 16] = √17

BC = √[(4-6)² + (8-0)²] = √[4 + 64] = √68 = 2√17

Conclusion: MN || BC and MN = ½ · BC (since √17 = ½ · 2√17)

Example 5: Proving Points are Collinear

Problem: Prove that A(1,2), B(3,6), and C(5,10) are collinear.

Solution:

Points are collinear if they all lie on the same line (same slope between all pairs).

Step 1: Find slope from A to B

m_AB = (6-2)/(3-1) = 4/2 = 2

Step 2: Find slope from B to C

m_BC = (10-6)/(5-3) = 4/2 = 2

Step 3: Find slope from A to C

m_AC = (10-2)/(5-1) = 8/4 = 2

All slopes equal 2.

Conclusion: A, B, and C are collinear because the slope between any two points is the same (2).

✏️ Practice

Try these problems on your own. Choose the best answer for each question.

1. Which property would you use slopes to prove?

A) Two segments have equal length

B) Two lines are parallel

C) A point is the midpoint of a segment

D) A triangle has a specific perimeter

2. If line AB has slope 3/4, what is the slope of a line perpendicular to AB?

A) 3/4

B) -3/4

C) 4/3

D) -4/3

3. ABCD has A(0,0), B(5,0), C(8,4), D(3,4). What type of quadrilateral is ABCD?

A) Rectangle

B) Parallelogram

C) Trapezoid

D) Square

4. To prove a quadrilateral is a rectangle, you should show that:

A) All sides are equal

B) Opposite sides are parallel and all angles are right angles

C) Diagonals bisect each other

D) It has exactly one pair of parallel sides

5. Points P(2,3), Q(6,7), R(x,y) are collinear with slope 1. If x = 10, what is y?

A) 11

B) 10

C) 14

D) 7

6. Triangle ABC has A(0,0), B(8,0), C(4,6). What is the best first step to prove it's isosceles?

A) Find all three slopes

B) Find the midpoint of each side

C) Calculate the length of each side

D) Find the equation of each side

7. Two lines have slopes m₁ = 2 and m₂ = -1/2. These lines are:

A) Parallel

B) Perpendicular

C) Neither parallel nor perpendicular

D) The same line

8. Which placement is best for proving properties of an isosceles right triangle?

A) Vertices at (0,0), (0,a), (a,a)

B) Vertices at (0,0), (a,0), (0,a)

C) Vertices at (1,1), (2,1), (1,2)

D) Vertices at (a,b), (c,d), (e,f)

9. To prove EFGH is a rhombus, you should show:

A) All four sides are equal in length

B) Diagonals are perpendicular

C) All angles are 90°

D) Either A or B

10. The slope of line segment from (a, 2a) to (3a, 4a) is:

A) 2

B) 1

C) a

D) 2a

Click to reveal answers
  1. B) Two lines are parallel — Parallel lines have equal slopes
  2. D) -4/3 — Perpendicular slopes are negative reciprocals: 3/4 → -4/3
  3. B) Parallelogram — AB || DC (both horizontal, slope 0) and AD || BC (both slope 4/3)
  4. B) Opposite sides are parallel and all angles are right angles — This defines a rectangle
  5. A) 11 — Using slope 1: (y-3)/(10-2) = 1, so y-3 = 8, y = 11
  6. C) Calculate the length of each side — Isosceles means at least two sides equal
  7. B) Perpendicular — m₁ × m₂ = 2 × (-1/2) = -1, confirming perpendicularity
  8. B) Vertices at (0,0), (a,0), (0,a) — Right angle at origin, equal legs of length a
  9. D) Either A or B — Both are valid ways to prove a rhombus
  10. B) 1 — m = (4a-2a)/(3a-a) = 2a/2a = 1

✅ Check Your Understanding

Question 1: Why is it advantageous to place one vertex of a figure at the origin?

Reveal Answer

Placing a vertex at the origin (0, 0) simplifies calculations significantly. When using the distance formula from the origin, it becomes √(x² + y²) instead of √[(x₂-x₁)² + (y₂-y₁)²]. When finding slopes from the origin, the formula simplifies to y/x. The midpoint formula also simplifies. This strategic placement reduces the number of terms to calculate and minimizes opportunities for arithmetic errors.

Question 2: What is the relationship between the slopes of perpendicular lines, and why does it work?

Reveal Answer

Perpendicular lines have slopes that are negative reciprocals: if one line has slope m, the perpendicular line has slope -1/m. This means m₁ × m₂ = -1. This relationship comes from the geometry of right angles. When a line rotates 90°, its "rise over run" essentially swaps and changes sign. A line going "up 2, right 3" (slope 2/3) when rotated 90° goes "up 3, left 2" (slope -3/2). The exception is horizontal and vertical lines, where one slope is 0 and the other is undefined.

Question 3: What are TWO different ways to prove that a quadrilateral is a parallelogram using coordinate geometry?

Reveal Answer

There are several methods: (1) Show both pairs of opposite sides are parallel by proving their slopes are equal; (2) Show both pairs of opposite sides are equal in length using the distance formula; (3) Show that the diagonals bisect each other by proving they have the same midpoint; (4) Show that one pair of opposite sides is both parallel and equal in length. Any of these methods is sufficient to prove a quadrilateral is a parallelogram.

Question 4: When using variables like (a, 0) instead of specific numbers like (3, 0), what advantage does this provide in a coordinate proof?

Reveal Answer

Using variables creates a general proof that works for ALL figures of that type, not just one specific example. When you prove a property using A(0,0), B(a,0), C(a,b), D(0,b) for a rectangle, you've proven it for every rectangle, regardless of its specific dimensions. This makes the proof more powerful and mathematically rigorous. It also reveals relationships between quantities that might not be obvious with specific numbers.

🚀 Next Steps

  • Review any concepts that felt challenging
  • Move on to the next lesson when ready
  • Return to practice problems periodically for review