Geometric Proofs Using Coordinates

A coordinate proof uses points on a coordinate plane together with algebra to prove that a geometric statement is true.

Instead of relying only on how a diagram looks, we calculate quantities such as:

• lengths
• slopes
• midpoints
• distances
• relationships between lines

These calculations provide mathematical evidence for conclusions such as:

• two segments are congruent
• two lines are parallel
• two lines are perpendicular
• a point is the midpoint of a segment
• a quadrilateral is a rectangle, parallelogram, rhombus, or square

Coordinate proofs combine geometry, algebra, and logical reasoning.

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Why Use Coordinates in Geometry?

A diagram can suggest that two sides are equal or that two lines are perpendicular, but appearance alone is not proof.

For example, two lines might look parallel on a drawing but have slightly different slopes.

Coordinate geometry allows us to replace visual estimates with calculations.

We can use:

distance → prove lengths are equal

slope → prove lines are parallel or perpendicular

midpoint → prove segments bisect each other

This creates a logical argument supported by numerical evidence.

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The Main Tools of Coordinate Proof

Three formulas are especially important.

Distance formula:

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

Slope formula:

m = (y₂ − y₁)/(x₂ − x₁)

Midpoint formula:

M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

Each formula provides a different type of geometric evidence.

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Choosing the Correct Tool

Before calculating anything, identify what you are trying to prove.

A strong coordinate proof calculates only what is necessary.

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Proving Segments Are Congruent

Two segments are congruent if they have the same length.

Suppose:

A(1, 2)

B(5, 5)

C(−2, 1)

D(2, 4)

We want to determine whether AB and CD are congruent.

For AB:

AB = √[(5 − 1)² + (5 − 2)²]

AB = √(4² + 3²)

AB = √25

AB = 5

For CD:

CD = √[(2 − (−2))² + (4 − 1)²]

CD = √(4² + 3²)

CD = √25

CD = 5

Therefore:

AB = CD

so:

AB and CD are congruent.

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Why Equal Distances Prove Congruence

Congruent line segments are defined as segments having equal lengths.

Therefore, once the distance formula establishes:

AB = CD

we have directly satisfied the definition of congruent segments.

The conclusion follows logically from the calculation.

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Using Squared Distances

Sometimes we do not need to calculate the square root.

Suppose:

AB² = 52

and:

CD² = 52

Since both segment lengths are positive:

AB = CD

Therefore, the segments are congruent.

This can make coordinate proofs faster.

For example, comparing:

√52 and √52

adds no useful information beyond recognizing that their squared lengths are equal.

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Proving a Triangle Is Isosceles

Suppose:

A(0, 0)

B(6, 0)

C(3, 4)

Find AC:

AC = √[(3 − 0)² + (4 − 0)²]

AC = √(9 + 16)

AC = 5

Find BC:

BC = √[(3 − 6)² + (4 − 0)²]

BC = √(9 + 16)

BC = 5

Therefore:

AC = BC

The triangle has two congruent sides.

Therefore:

triangle ABC is isosceles.

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Proving a Triangle Is Equilateral

To prove a triangle is equilateral, show that:

AB = BC = CA

using the distance formula.

All three sides must have equal length.

Showing only two equal sides proves that the triangle is isosceles, but it does not by itself prove that it is equilateral.

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Using Slope to Prove Lines Are Parallel

For nonvertical lines:

equal slopes indicate parallel direction

Suppose:

A(1, 2)

B(5, 6)

C(−2, 3)

D(2, 7)

Find the slope of AB:

m_AB = (6 − 2)/(5 − 1)

m_AB = 4/4

m_AB = 1

Find the slope of CD:

m_CD = (7 − 3)/(2 − (−2))

m_CD = 4/4

m_CD = 1

The slopes are equal.

Therefore, AB and CD have the same direction.

If they are distinct lines:

AB ∥ CD

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Equal Slopes Need One Important Check

Suppose two segments have equal slopes.

This shows that their supporting lines have the same direction.

However, they could theoretically lie on the same line.

Therefore, when proving that two full lines are distinct parallel lines, we should also know that they are not the same line.

In most quadrilateral coordinate proofs, the diagram and distinct vertices make this clear.

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Vertical Parallel Lines

The ordinary slope formula does not produce a numerical slope for vertical lines.

For example:

AB joins:

A(2, 1)

B(2, 7)

Both points have:

x = 2

Therefore AB is vertical.

CD joins:

C(6, −2)

D(6, 5)

Both points have:

x = 6

Therefore CD is also vertical.

The lines are distinct and vertical.

Therefore:

AB ∥ CD

We should not try to assign them a numerical slope.

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Proving Lines Are Perpendicular

For two nonvertical, nonhorizontal lines, perpendicular slopes are negative reciprocals.

Equivalently:

m₁m₂ = −1

Suppose:

A(1, 1)

B(5, 3)

C(2, 6)

D(3, 4)

Slope AB:

m_AB = (3 − 1)/(5 − 1)

m_AB = 2/4

m_AB = 1/2

Slope CD:

m_CD = (4 − 6)/(3 − 2)

m_CD = −2/1

m_CD = −2

Multiply:

(1/2)(−2) = −1

Therefore:

AB ⟂ CD

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Horizontal and Vertical Lines

A horizontal line has:

slope = 0

A vertical line has:

undefined slope

Horizontal and vertical lines are perpendicular.

For example:

A(1, 3)

B(7, 3)

creates a horizontal segment because the y-coordinate remains constant.

C(4, −2)

D(4, 6)

creates a vertical segment because the x-coordinate remains constant.

Therefore:

AB ⟂ CD

This is a special case where the negative-reciprocal calculation is not used.

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Proving a Right Angle

Suppose we want to prove that angle ABC is 90°.

We calculate:

• slope of BA
• slope of BC

If the slopes are negative reciprocals, then:

BA ⟂ BC

Therefore:

∠ABC = 90°

Coordinate geometry allows an angle relationship to be proved using line slopes.

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Worked Right-Triangle Proof

Consider:

A(0, 0)

B(6, 0)

C(6, 4)

Slope AB:

m_AB = 0

BC is vertical because both B and C have:

x = 6

Therefore:

AB ⟂ BC

Thus:

∠ABC = 90°

and:

triangle ABC is a right triangle.

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The Midpoint Formula

The midpoint of a segment lies exactly halfway between its endpoints.

For:

A(x₁, y₁)

B(x₂, y₂)

the midpoint is:

M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

For example:

A(2, 3)

B(8, 7)

gives:

M = ((2 + 8)/2, (3 + 7)/2)

M = (10/2, 10/2)

M = (5, 5)

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Using Midpoints in Proofs

Midpoints can prove that segments:

• bisect each other
• share the same center
• divide a figure symmetrically

Suppose two diagonals have the same midpoint.

Then each diagonal passes through the midpoint of the other.

Therefore:

the diagonals bisect each other

This is especially useful when proving that a quadrilateral is a parallelogram.

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Proving Diagonals Bisect Each Other

Suppose quadrilateral ABCD has:

A(0, 0)

B(6, 0)

C(8, 4)

D(2, 4)

Its diagonals are AC and BD.

Midpoint of AC:

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

M_AC = (4, 2)

Midpoint of BD:

M_BD = ((6 + 2)/2, (0 + 4)/2)

M_BD = (4, 2)

The diagonals have the same midpoint.

Therefore:

the diagonals bisect each other

A quadrilateral whose diagonals bisect each other is a parallelogram.

Therefore:

ABCD is a parallelogram.

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More Than One Way to Prove a Parallelogram

Coordinate geometry often provides several possible proofs.

A quadrilateral can be proved to be a parallelogram by showing that:

• both pairs of opposite sides are parallel
• both pairs of opposite sides are congruent
• one pair of opposite sides is both parallel and congruent
• the diagonals bisect each other

The best method depends on the coordinates.

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Proving a Parallelogram Using Slopes

Consider:

A(1, 1)

B(6, 1)

C(8, 4)

D(3, 4)

Slope AB:

0

Slope CD:

0

Therefore:

AB ∥ CD

Slope BC:

m_BC = (4 − 1)/(8 − 6)

m_BC = 3/2

Slope AD:

m_AD = (4 − 1)/(3 − 1)

m_AD = 3/2

Therefore:

BC ∥ AD

Both pairs of opposite sides are parallel.

Therefore:

ABCD is a parallelogram.

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Proving a Rectangle

A rectangle is a quadrilateral with four right angles.

One efficient coordinate proof is to:

1. prove the quadrilateral is a parallelogram
2. prove one pair of adjacent sides is perpendicular

A parallelogram with one right angle must have four right angles.

Therefore, it is a rectangle.

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Rectangle Example

Suppose:

A(1, 1)

B(7, 1)

C(7, 5)

D(1, 5)

AB and CD are horizontal.

BC and AD are vertical.

Therefore:

AB ∥ CD

BC ∥ AD

So the quadrilateral is a parallelogram.

Also:

AB ⟂ BC

Therefore, it contains a right angle.

Thus:

ABCD is a rectangle.

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Proving a Rhombus

A rhombus has four congruent sides.

To prove a quadrilateral is a rhombus using coordinates, calculate the lengths of all four sides.

If:

AB = BC = CD = DA

then the quadrilateral is a rhombus.

Using squared distances can make this calculation faster.

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Rhombus Example

Consider:

A(0, 3)

B(4, 0)

C(8, 3)

D(4, 6)

Calculate squared side lengths.

AB²:

(4 − 0)² + (0 − 3)²

= 16 + 9

= 25

BC²:

(8 − 4)² + (3 − 0)²

= 16 + 9

= 25

CD²:

(4 − 8)² + (6 − 3)²

= 16 + 9

= 25

DA²:

(0 − 4)² + (3 − 6)²

= 16 + 9

= 25

All four side lengths are equal.

Therefore:

ABCD is a rhombus.

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Proving a Square

A square has:

• four congruent sides
• four right angles

There are several valid coordinate-proof strategies.

One method is:

1. prove all four sides are congruent
2. prove one pair of adjacent sides is perpendicular

This proves the figure is both a rhombus and a rectangle.

Therefore:

the quadrilateral is a square

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Square Example

Suppose:

A(0, 0)

B(4, 0)

C(4, 4)

D(0, 4)

Side lengths:

AB = 4

BC = 4

CD = 4

DA = 4

Therefore, all sides are congruent.

AB is horizontal.

BC is vertical.

Therefore:

AB ⟂ BC

The quadrilateral has four congruent sides and a right angle.

Therefore:

ABCD is a square.

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Proving a Shape Is Not What It Appears to Be

Coordinate proofs can also disprove claims.

Suppose someone claims that a quadrilateral is a rectangle.

If adjacent slopes are:

m₁ = 2

m₂ = −1/3

then:

m₁m₂ = 2(−1/3)

= −2/3

Since:

−2/3 ≠ −1

the lines are not perpendicular.

Therefore, that angle is not 90°.

The figure cannot be a rectangle based on those sides.

This shows why visual appearance is not sufficient.

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Proving an Isosceles Right Triangle

Sometimes more than one property must be established.

To prove a triangle is an isosceles right triangle, we need to show:

• two sides are congruent
• one angle is 90°

Suppose:

A(0, 0)

B(4, 0)

C(0, 4)

Lengths:

AB = 4

AC = 4

Therefore:

AB = AC

So the triangle is isosceles.

AB is horizontal.

AC is vertical.

Therefore:

AB ⟂ AC

Thus ∠A = 90°.

Therefore:

triangle ABC is an isosceles right triangle.

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Using Midpoints to Prove a Segment Is Bisected

Suppose:

A(−2, 4)

B(8, 10)

and point M is:

(3, 7)

Find the midpoint of AB:

M_AB = ((−2 + 8)/2, (4 + 10)/2)

M_AB = (6/2, 14/2)

M_AB = (3, 7)

This is exactly point M.

Therefore:

M is the midpoint of AB

and:

M bisects AB

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Using Both Distance and Midpoint

Sometimes a midpoint claim can also be supported using distances.

If M lies on AB and:

AM = MB

then M bisects AB.

For coordinate proofs, however, the midpoint formula is usually the more direct method.

Good proofs choose the simplest tool that establishes the required fact.

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Proving a Perpendicular Bisector

To prove that a line is the perpendicular bisector of a segment, we need two facts:

1. it passes through the segment's midpoint
2. it is perpendicular to the segment

Suppose segment AB has endpoints:

A(2, 2)

B(8, 6)

First find the midpoint:

M = ((2 + 8)/2, (2 + 6)/2)

M = (5, 4)

Slope of AB:

m_AB = (6 − 2)/(8 − 2)

m_AB = 4/6

m_AB = 2/3

Therefore, a perpendicular line has slope:

−3/2

Any line passing through (5, 4) with slope −3/2 is the perpendicular bisector of AB.

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Why Perpendicular Bisectors Matter

Every point on the perpendicular bisector of segment AB is equally distant from A and B.

Therefore, if point P lies on the perpendicular bisector:

PA = PB

This property is important in:

• triangle geometry
• circle construction
• locating centers
• surveying
• design

For example, the center of a circle lies on the perpendicular bisector of every chord.

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Proving a Point Is Equidistant from Two Points

Suppose:

A(1, 2)

B(7, 2)

P(4, 6)

Find PA²:

PA² = (4 − 1)² + (6 − 2)²

= 3² + 4²

= 25

Find PB²:

PB² = (4 − 7)² + (6 − 2)²

= (−3)² + 4²

= 25

Therefore:

PA = PB

Point P is equidistant from A and B.

This provides coordinate evidence that P lies on the perpendicular bisector of AB.

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Coordinate Proofs and Circles

Coordinate reasoning can also prove properties involving circles.

Suppose a circle has center:

O(2, 3)

and points:

A(5, 7)

B(−1, 7)

Find OA:

OA = √[(5 − 2)² + (7 − 3)²]

OA = √(9 + 16)

OA = 5

Find OB:

OB = √[(−1 − 2)² + (7 − 3)²]

OB = √(9 + 16)

OB = 5

Therefore:

OA = OB

Both points are the same distance from O.

If the circle has radius 5, both A and B lie on the circle.

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Proving a Line Is Tangent to a Circle

Suppose a circle has center O and point T lies on the circle.

To prove a line through T is tangent, show that:

radius OT ⟂ the line

In coordinate geometry:

1. find the slope of OT
2. find the slope of the proposed tangent
3. show that the slopes are negative reciprocals

Then the lines are perpendicular.

By the radius-tangent theorem, the line is tangent to the circle.

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Tangent Proof Example

Suppose a circle has center:

O(0, 0)

and T is:

(3, 4)

The radius slope is:

m_OT = 4/3

Suppose a line through T has slope:

−3/4

Then:

(4/3)(−3/4) = −1

Therefore:

OT ⟂ tangent line

Since T lies on the circle and the line through T is perpendicular to the radius:

the line is tangent to the circle at T.

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Constructing a Coordinate Proof

A coordinate proof should follow a logical sequence.

A useful structure is:

Given: State the coordinates and relevant information.

Prove: State the geometric property to be established.

Calculate: Use distance, slope, midpoint, or another appropriate relationship.

Interpret: Explain what the calculations mean geometrically.

Conclude: State clearly what has been proved.

The calculations alone are not the entire proof. The reasoning connecting the calculations to the conclusion is essential.

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Example of a Complete Coordinate Proof

Given:

A(0, 0)

B(6, 0)

C(8, 4)

D(2, 4)

Prove: ABCD is a parallelogram.

Calculate slope AB:

m_AB = (0 − 0)/(6 − 0)

m_AB = 0

Calculate slope CD:

m_CD = (4 − 4)/(2 − 8)

m_CD = 0

Therefore:

AB ∥ CD

Now calculate slope BC:

m_BC = (4 − 0)/(8 − 6)

m_BC = 4/2

m_BC = 2

Calculate slope AD:

m_AD = (4 − 0)/(2 − 0)

m_AD = 4/2

m_AD = 2

Therefore:

BC ∥ AD

Both pairs of opposite sides are parallel.

Therefore:

ABCD is a parallelogram.

This is a complete argument because every calculation is connected to a geometric property.

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Another Proof of the Same Parallelogram

We could instead use the diagonals.

Diagonal AC has midpoint:

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

= (4, 2)

Diagonal BD has midpoint:

((6 + 2)/2, (0 + 4)/2)

= (4, 2)

The diagonals have the same midpoint.

Therefore:

the diagonals bisect each other

A quadrilateral whose diagonals bisect each other is a parallelogram.

Therefore:

ABCD is a parallelogram.

This illustrates an important feature of mathematics:

there can be more than one valid proof.

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Efficient Proofs

A good coordinate proof does not necessarily calculate everything possible.

Suppose you need to prove that a quadrilateral is a parallelogram.

You do not need to calculate:

• all four side lengths
• all four slopes
• both diagonal lengths
• both diagonal midpoints

if one simpler argument already proves the result.

For example, showing that the diagonals have the same midpoint may be enough.

Efficiency is part of good mathematical reasoning.

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Choosing Convenient Coordinates

When creating your own coordinate proof, choosing convenient coordinates can simplify the mathematics.

For example, a rectangle might be placed at:

A(0, 0)

B(a, 0)

C(a, b)

D(0, b)

Now:

• AB and CD are horizontal
• BC and AD are vertical
• perpendicularity is obvious from coordinate relationships
• distances are easy to calculate

Good coordinate placement can greatly reduce unnecessary algebra.

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Using Variables in Coordinate Proofs

Coordinate proofs do not always require numerical coordinates.

Suppose a rectangle has:

A(0, 0)

B(a, 0)

C(a, b)

D(0, b)

AB is horizontal because both endpoints have y = 0.

CD is horizontal because both endpoints have y = b.

Therefore:

AB ∥ CD

BC is vertical because both endpoints have x = a.

AD is vertical because both endpoints have x = 0.

Therefore:

BC ∥ AD

Horizontal and vertical lines are perpendicular.

Therefore:

AB ⟂ BC

Thus ABCD has opposite sides parallel and a right angle.

Therefore:

ABCD is a rectangle.

This proves the property for any suitable positive values of a and b.

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General Proofs vs Specific Examples

A proof using numerical coordinates proves a property for that particular figure.

For example:

A(0, 0), B(4, 0), C(4, 3), D(0, 3)

proves that this particular quadrilateral is a rectangle.

A proof using variables such as:

A(0, 0), B(a, 0), C(a, b), D(0, b)

can establish a more general result.

This distinction is important in mathematical reasoning.

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Logical Reasoning in Coordinate Proofs

A strong proof follows a chain such as:

Calculation → geometric fact → theorem or definition → conclusion

For example:

m_AB = 2

m_CD = 2

↓

equal slopes

↓

AB ∥ CD

↓

opposite sides are parallel

↓

supports proof that ABCD is a parallelogram

Each step should have a mathematical reason.

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Coordinate Proofs in Real-World Design

Coordinate methods are useful beyond classroom geometry.

Applications include:

• computer-aided design
• architecture
• engineering drawings
• surveying
• mapping
• robotics
• computer graphics
• manufacturing

Designers often represent important points using coordinates and then verify dimensions and relationships mathematically.

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Engineering Example

Suppose an engineer needs two structural supports to be parallel.

Support A connects:

(2, 3) to (8, 6)

Support B connects:

(4, 8) to (10, 11)

Slope A:

(6 − 3)/(8 − 2)

= 3/6

= 1/2

Slope B:

(11 − 8)/(10 − 4)

= 3/6

= 1/2

The slopes are equal.

Therefore, the supports are mathematically parallel in the coordinate model.

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Computer Graphics

Computer graphics represents objects using coordinate points.

Software can use:

• distances to preserve dimensions
• slopes and vectors to control direction
• midpoints to position objects
• perpendicular relationships to construct corners
• geometric constraints to maintain shapes

Coordinate proofs use many of the same mathematical ideas.

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A Reliable Strategy for Coordinate Proofs

When constructing a coordinate proof:

1. Identify exactly what must be proved.
2. Decide which geometric property would establish the conclusion.
3. Select the appropriate coordinate tool.
4. Label the relevant points and segments.
5. Calculate carefully.
6. Keep subtraction order consistent.
7. Simplify exact values when possible.
8. Compare the results.
9. State the geometric relationship established by the calculation.
10. Connect that relationship to a definition or theorem.
11. State the final conclusion clearly.

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Distance, Slope, or Midpoint?

Ask yourself:

Is the problem about equal lengths?

Use distance.

Is it about parallel or perpendicular lines?

Use slope.

Is it about bisecting segments or diagonals?

Use midpoint.

Does the figure require several properties?

Use a combination.

The goal is not to use every formula. The goal is to produce sufficient evidence for the conclusion.

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Common Mistakes

Relying on appearance

A diagram is evidence for what to investigate, not proof.

Using distance when slope is needed

Equal lengths do not prove that lines are parallel.

Using slope when distance is needed

Parallel sides do not necessarily have equal lengths.

Forgetting vertical-line cases

Vertical lines have undefined slope. Two distinct vertical lines are parallel, and a vertical line is perpendicular to a horizontal line.

Changing only the sign for perpendicular slopes

The perpendicular slope to 2/3 is:

−3/2

not −2/3.

Using inconsistent subtraction in the slope formula

If the numerator is y₂ − y₁, the denominator must be x₂ − x₁.

Calculating only one side when proving a rhombus

A rhombus requires four congruent sides.

Showing equal sides but claiming a square

Equal sides establish a rhombus. A square also requires right angles.

Showing perpendicular sides but claiming a rectangle

A single right angle alone does not prove an arbitrary quadrilateral is a rectangle. Additional information is required.

Finding equal diagonal lengths and claiming a parallelogram

Equal diagonals alone do not guarantee that a quadrilateral is a parallelogram.

Finding the same diagonal midpoint without explaining its meaning

State explicitly that the diagonals bisect each other and then use the appropriate theorem.

Stopping after the calculations

A proof needs a conclusion explaining why the calculations establish the required geometric property.

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Key Terms

Coordinate proof — proof using coordinates, algebraic calculations, and geometric definitions or theorems.

Distance formula — formula used to calculate the length between two coordinate points.

Slope — measure of a line's rate of change and direction.

Midpoint — point exactly halfway between two endpoints.

Congruent segments — segments having equal lengths.

Parallel lines — coplanar lines that never intersect; distinct nonvertical parallel lines have equal slopes.

Perpendicular lines — lines meeting at a 90° angle.

Negative reciprocal — relationship between the slopes of nonvertical, nonhorizontal perpendicular lines.

Bisect — divide into two equal parts.

Perpendicular bisector — line perpendicular to a segment at its midpoint.

Equidistant — located the same distance from two or more points.

Parallelogram — quadrilateral with both pairs of opposite sides parallel.

Rectangle — parallelogram with four right angles.

Rhombus — parallelogram with four congruent sides.

Square — quadrilateral with four congruent sides and four right angles.

Logical reasoning — connecting mathematical evidence to conclusions through valid definitions and theorems.

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Key Takeaways

• Coordinate proofs use algebra to establish geometric properties.
• A diagram alone is not sufficient proof.
• The distance formula can prove that segments are congruent.
• Comparing squared distances can often simplify length proofs.
• Equal slopes can prove that distinct nonvertical lines are parallel.
• Negative reciprocal slopes can prove that nonvertical, nonhorizontal lines are perpendicular.
• Horizontal and vertical lines form an important perpendicular special case.
• The midpoint formula can prove that a segment is bisected.
• If the diagonals of a quadrilateral have the same midpoint, they bisect each other.
• Diagonals that bisect each other can be used to prove a quadrilateral is a parallelogram.
• Distance can help prove that a triangle is isosceles or that a quadrilateral is a rhombus.
• Slope can help prove right angles, rectangles, and parallel sides.
• A combination of distance and slope can prove that a figure is a square.
• A perpendicular bisector requires both midpoint and perpendicularity.
• Coordinate methods can prove circle properties such as equal radii and tangent relationships.
• More than one valid coordinate proof may exist for the same figure.
• Efficient proofs calculate only the information necessary to establish the conclusion.
• A complete proof connects calculations to geometric definitions and theorems.
• Coordinate proofs are widely applicable in engineering, design, surveying, robotics, and computer graphics.

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Check Your Understanding

1. Explain why a drawing that appears to show two equal segments is not sufficient to prove that the segments are congruent.
2. Find the lengths of AB and CD for A(1, 2), B(5, 5), C(3, −1), and D(7, 2). Determine whether the segments are congruent.
3. Use slopes to determine whether the line through A(1, 3) and B(5, 7) is parallel to the line through C(−2, 4) and D(2, 8).
4. Determine whether the lines through A(0, 0), B(6, 3) and C(2, 7), D(4, 3) are perpendicular.
5. Find the midpoint of the segment joining (−4, 6) and (8, −2).
6. Point M is (3, 5). Determine whether M is the midpoint of the segment joining A(−1, 2) and B(7, 8).
7. Triangle ABC has vertices A(0, 0), B(6, 0), and C(3, 4). Use coordinate geometry to prove that the triangle is isosceles.
8. Triangle ABC has vertices A(0, 0), B(5, 0), and C(0, 5). Prove that it is an isosceles right triangle.
9. Quadrilateral ABCD has vertices A(0, 0), B(6, 0), C(8, 4), and D(2, 4). Use slopes to prove that it is a parallelogram.
10. Prove the same quadrilateral in Question 9 is a parallelogram using the midpoints of its diagonals instead of slopes.
11. Quadrilateral ABCD has vertices A(1, 1), B(7, 1), C(7, 5), and D(1, 5). Use coordinate relationships to prove that it is a rectangle.
12. Quadrilateral ABCD has vertices A(0, 3), B(4, 0), C(8, 3), and D(4, 6). Use distances to prove that it is a rhombus.
13. Explain what two types of coordinate evidence could be used together to prove that a quadrilateral is a square.
14. A circle has center O(0, 0) and passes through T(3, 4). A line through T has slope −3/4. Use coordinate reasoning to prove that the line is tangent to the circle at T.
15. A student calculates the slopes of two opposite sides of a quadrilateral and finds that both slopes equal 2. The student immediately concludes that the quadrilateral is a parallelogram. Explain why this is not enough evidence and describe what additional coordinate calculation could complete the proof.