Geometry Workbook — Chapter 11: Quadrilaterals in the Coordinate Plane

SOL G.PC.1 (a, b) · Companion to Textbook Chapter 11

Each page below is one Canva page. Headings are sized for direct paste: page title as H1, section labels as H2. Figures referenced by filename live in ../figures/. Every numbered item is the same problem as in the textbook, so one answer key serves both. Item numbers run continuously from 1 to 112.

PAGE 1 — Chapter opener

Chapter 11 · Quadrilaterals in the Coordinate Plane

Standard G.PC.1 (a, b)

In this chapter you will:

Say which of the three formulas settles a given claim
Prove a parallelogram four ways, and pick the cheapest
Prove a rectangle, a rhombus, and a square from coordinates
Prove a trapezoid — including the step most people skip
Choose a proof instead of computing everything in sight
Run the formulas backwards to find a missing coordinate

Words to know: coordinate plane · slope formula · distance formula · midpoint formula · parallel · perpendicular · congruent · bisect · simplest radical form · coordinate proof · converse · most specific family

Conventions: name the formula before you compute. Answers exact, in simplest radical form — never compare two decimals. A picture is not a proof. A trapezoid has exactly one pair of parallel sides, so its proof needs two slope facts.

PAGE 2 — The three formulas

11.1 One Tool for Each Kind of Claim

FIGURE: fig1-the-three-formulas.png (full width)

Fill in the right-hand column.

Formula	What you compute	What it proves
slope	$m = \dfrac{y_2-y_1}{x_2-x_1}$	____________________
distance	$d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$	____________________
midpoint	$M = \left(\dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2}\right)$	____________________

Which formula proves two segments are congruent? ____________
Which proves two segments are parallel? ____________ What must be true? ____________
Which proves the diagonals bisect each other? ____________

Learn the table right to left. You will not be asked for a slope — you will be asked whether something is a parallelogram, and the work is realising that is a parallel question.

PAGE 3 — Slope proves two things

Parallel, and Perpendicular

FIGURE: fig2-slope-proves-parallel-and-perpendicular.png (full width)

Left panel: the two slopes are ______ and ______ , which proves ____________________
Right panel: the two slopes are ______ and ______ , their product is ______ , which proves ____________________

A vertical segment has an undefined slope, not zero. You cannot prove perpendicular by multiplying when one slope is undefined. Say instead: one is vertical and one is horizontal.

PAGE 4 — Distance and midpoint

Congruent, and Bisects

FIGURE: fig3-distance-proves-congruent.png (half width) FIGURE: fig4-midpoint-proves-bisects.png (half width)

On the midpoint figure, $M =$ ________ Why does one point prove something about two segments?

Leave the radical exact. $\sqrt{40} =$ ____________ , and that is what you write.

PAGE 5 — Practice · run the formulas

Practice

Find each, exactly.

Slope of $\overline{AB}$ , $A(-4,1)$ , $B(0,-3)$ . ______
Slope of $\overline{CD}$ , $C(6,3)$ , $D(2,7)$ . ______
$AB$ for $A(-4,-3)$ , $B(4,-3)$ . ______
$CD$ for $C(2,3)$ , $D(-2,3)$ . ______
$AD$ for $A(-4,-3)$ , $D(-2,3)$ , in simplest radical form. ______
Midpoint of $\overline{AC}$ , $A(0,0)$ , $C(7,6)$ . ______
Midpoint of $\overline{BD}$ , $B(6,2)$ , $D(1,4)$ . ______
Slope of a vertical segment: ______ Of a horizontal one: ______

PAGE 6 — Practice · name the formula only

Which Formula? (Do Not Compute)

Show one pair of opposite sides is congruent. ____________
Show $\angle A$ is a right angle. ____________
Show $\overline{AC}$ and $\overline{BD}$ bisect each other. ____________
Show $\overline{AD} \parallel \overline{BC}$ . ____________
Application. Stakes at $A(-4,-3)$ , $B(4,-3)$ , $C(2,3)$ , $D(-2,3)$ . Do the north and south edges run parallel?

Formula: ____________ Work: ____________________ Answer: ______

PAGE 7 — Think it through · 11.1

Think It Through

Error analysis. A student writes $AB = \sqrt{40} \approx 6.3$ and $CD = \sqrt{40} \approx 6.3$ , then concludes " $AB \approx CD$ ." Why is the conclusion weaker than what they proved?
Reasoning. Why can slope certify a $90°$ angle but no other angle measure?
Reasoning. A student says "the midpoint came out to $\left(\tfrac72, 3\right)$ , so I made a mistake." Correct them.

Exit ticket 11.1

Which formula proves perpendicular? ____________ The two numbers must ____________
Slope of $\overline{AB}$ , $A(2,1)$ , $B(5,2)$ : ______
$AB$ for those points: ______
Midpoint of $\overline{AB}$ for those points: ______

PAGE 8 — Four routes to a parallelogram

11.2 Four Ways, All Valid

FIGURE: fig6-three-routes-to-a-parallelogram.png (full width)

Name the four routes.

· ____________________ · ____________________

· ____________________ · ____________________
Which takes the fewest computations? ____________ How many? ______
Why is "one pair of opposite sides parallel" not on the list by itself?

PAGE 9 — The slope route

Four Slopes, Two Pairs

FIGURE: fig5-parallelogram-by-slope.png (full width)

Give the four slopes and say what pairing them proves.

$m_{AB} =$ ____ $m_{DC} =$ ____ $m_{AD} =$ ____ $m_{BC} =$ ____

Conclusion: ____________________
Give one reason it is not a rectangle and one reason it is not a rhombus.

PAGE 10 — Writing the proof

The Three Parts of a Coordinate Proof

Every coordinate proof has a computation, a comparison, and a conclusion that names the property. All three go on the page.

Model. Given $A(-6,-2)$ , $B(0,-4)$ , $C(4,2)$ , $D(-2,4)$ . Prove $ABCD$ is a parallelogram.

Midpoint of $\overline{AC} = \left(\dfrac{-6+4}{2}, \dfrac{-2+2}{2}\right) = (-1,0)$ Midpoint of $\overline{BD} = \left(\dfrac{0+(-2)}{2}, \dfrac{-4+4}{2}\right) = (-1,0)$ The midpoints are the same point, so $\overline{AC}$ and $\overline{BD}$ bisect each other. A quadrilateral whose diagonals bisect each other is a parallelogram. $\blacksquare$

The line people drop is the last one. Two matching midpoints are a computation, not a conclusion.

PAGE 11 — Solving for a missing vertex

Run the Formula Backwards

FIGURE: fig13-solve-for-the-missing-vertex.png (full width)

Which property locates $D$ ? ____________________

Why does it pin $D$ down exactly? ____________________

The method. $M$ is fixed by $A$ and $C$ . Then set the midpoint of $\overline{BD}$ equal to $M$ and solve:

$\frac{0+x}{2} = -1 \Rightarrow x = \underline{\phantom{-2}} \qquad \frac{-4+y}{2} = 0 \Rightarrow y = \underline{\phantom{4}}$

PAGE 12 — Practice · is it a parallelogram?

Decide, and Show the Work

$A(-5,1)$ , $B(-1,-2)$ , $C(5,2)$ , $D(1,5)$ — midpoints.

mid $\overline{AC} =$ ______ mid $\overline{BD} =$ ______ Answer: ______
$A(0,0)$ , $B(6,2)$ , $C(7,6)$ , $D(1,4)$ — slopes.

____ ____ ____ ____ Answer: ______
$A(-4,-3)$ , $B(4,-3)$ , $C(3,3)$ , $D(-1,3)$ — midpoints.

mid $\overline{AC} =$ ______ mid $\overline{BD} =$ ______ Answer: ______
$A(-3,-1)$ , $B(3,-3)$ , $C(5,3)$ , $D(-1,5)$ — midpoints.

mid $\overline{AC} =$ ______ mid $\overline{BD} =$ ______ Answer: ______

PAGE 13 — Practice · the arithmetic

Practice

Midpoint of $\overline{AC}$ , $A(-5,1)$ , $C(5,2)$ . ______
Midpoint of $\overline{BD}$ , $B(-1,-2)$ , $D(1,5)$ . ______
$m_{AB}$ and $m_{DC}$ for $A(0,0)$ , $B(6,2)$ , $C(7,6)$ , $D(1,4)$ . ______ ______
$AB$ and $DC$ for those points. ______ ______
Parallelogram $ABCD$ , $A(1,2)$ , $B(5,3)$ , $C(6,7)$ . $D =$ ______
Parallelogram $ABCD$ , $A(-6,-2)$ , $B(0,-4)$ , $C(4,2)$ . $D =$ ______
Diagonals meet at $M(2,-1)$ , $A(-3,4)$ . $C =$ ______
Diagonals meet at $M(0,3)$ , $B(4,6)$ . $D =$ ______

PAGE 14 — Practice · write the proofs

Write It Out

Full coordinate proof that $A(-5,1)$ , $B(-1,-2)$ , $C(5,2)$ , $D(1,5)$ is a parallelogram, by midpoints.
The same figure by slopes. Then say which proof you would rather write, and why.
Prove $A(0,0)$ , $B(6,2)$ , $C(7,6)$ , $D(1,4)$ is a parallelogram using one pair of opposite sides only.

PAGE 15 — Think it through · 11.2

Think It Through

Application. A parking bay is marked at $A(-5,1)$ , $B(-1,-2)$ , $C(5,2)$ , $D(1,5)$ , in metres. Verify that opposite edges are parallel, and give the two edge lengths.
Error analysis. A student shows $AB = CD$ and concludes $ABCD$ is a parallelogram. Give a reason it does not follow, and one extra fact that would fix it.
Error analysis. A student takes the midpoint of $\overline{AB}$ and of $\overline{CD}$ , finds them different, and concludes it is not a parallelogram. Identify the error.
Reasoning. Why does the midpoint route need two computations where the slope route needs four?
Reasoning. Why is "both pairs of opposite sides congruent" a theorem you may use rather than the definition?

Exit ticket 11.2

Cheapest parallelogram test: ____________________ How many? ______
Is $A(-3,-1)$ , $B(3,-3)$ , $C(5,3)$ , $D(-1,5)$ a parallelogram? ______ Midpoints: ______ ______
Parallelogram $ABCD$ , $A(0,0)$ , $B(6,2)$ , $C(7,6)$ . $D =$ ______
The line students most often leave out: ____________________

PAGE 16 — The ladder

11.3 Parallelogram First, Then One More Check

FIGURE: fig10-which-test-which-family.png (full width)

What do the rectangle, rhombus, and square rows all start with? ____________________
The one extra check for a rectangle: ____________________

The one extra check for a rhombus: ____________________

Two adjacent sides, not four. Opposite sides of a parallelogram are already congruent. So $AB = AD$ makes all four congruent, and computing $BC$ and $CD$ tells you nothing new.

PAGE 17 — A rectangle

Two Complete Routes

FIGURE: fig7-rectangle-in-the-plane.png (full width)

Slope route: $m_{AB} =$ ____ , $m_{AD} =$ ____ , product $=$ ____ , so ____________________

Distance route: $AC =$ ____ , $BD =$ ____ , so ____________________

What does not finish it: a right angle with no ____________________ established first.

PAGE 18 — A rhombus

Four Fives

FIGURE: fig8-rhombus-in-the-plane.png (full width)

Explain why the diagonals are perpendicular without multiplying two slopes.
$AC =$ ______ $BD =$ ______ Their being different rules out ____________________

PAGE 19 — A square

Both Extra Checks

FIGURE: fig9-square-in-the-plane.png (full width)

List the four facts in the order you would prove them.
1. ____________________ → ____________
2. ____________________ → ____________
3. ____________________ → ____________
4. ____________________ → ____________

Six computations. Computing all four sides and all four slopes and both diagonals is ten — for the same conclusion.

PAGE 20 — Practice · name the family

Most Specific Family, With Proof

$A(-4,1)$ , $B(0,-3)$ , $C(6,3)$ , $D(2,7)$ . ______________
$A(-6,0)$ , $B(0,-3)$ , $C(6,0)$ , $D(0,3)$ . ______________
$A(-3,-1)$ , $B(3,-3)$ , $C(5,3)$ , $D(-1,5)$ . ______________
$A(0,0)$ , $B(6,2)$ , $C(7,6)$ , $D(1,4)$ . ______________

PAGE 21 — Practice · the individual checks

Practice

$AC$ and $BD$ for $A(-4,1)$ , $B(0,-3)$ , $C(6,3)$ , $D(2,7)$ : ______ ______ Proves: ____________
$AB$ and $AD$ for those points: ______ ______ Rules out: ____________
$AB$ and $AD$ for $A(-6,0)$ , $B(0,-3)$ , $D(0,3)$ : ______ ______ Proves: ____________
Slopes of both diagonals of $A(1,0)$ , $B(5,3)$ , $C(1,6)$ , $D(-3,3)$ : ______ ______

State the conclusion carefully: ____________________
$AC$ and $BD$ for that rhombus: ______ ______ Rules out: ____________
$AB$ and $AD$ for $A(2,1)$ , $B(5,2)$ , $D(1,4)$ : ______ ______
$AC$ and $BD$ for $A(2,1)$ , $B(5,2)$ , $C(4,5)$ , $D(1,4)$ : ______ ______
A parallelogram with $AC = BD$ : ______________
A parallelogram with $AB = AD$ : ______________
A parallelogram with both: ______________

PAGE 22 — Think it through · 11.3

Think It Through

Application. A tile has corners $A(2,1)$ , $B(5,2)$ , $C(4,5)$ , $D(1,4)$ , in inches. Prove it is a square in the fewest computations you can, and give the side exactly.
Application. A gate frame has corners $A(-4,1)$ , $B(0,-3)$ , $C(6,3)$ , $D(2,7)$ . It must be square-cornered but not equal-sided. Verify both.
Error analysis. A student finds a parallelogram's four sides are $5,5,5,5$ and writes "so it is a square." Correct them.
Error analysis. A student shows $\angle A$ is right and concludes "rectangle," with nothing else shown. What is missing?
Reasoning. Why is checking two adjacent sides enough for a rhombus, once you know it is a parallelogram?
Reasoning. A rhombus has a vertical diagonal and a horizontal one. Why can you not multiply slopes, and what do you write instead?

Exit ticket 11.3

Parallelogram → rectangle needs: ____________________
Parallelogram → rhombus needs: ____________________
Is $A(-6,0)$ , $B(0,-3)$ , $C(6,0)$ , $D(0,3)$ a square? ______ Deciding computation: ____________
Fewest computations for a square: ______ They are: ____________________

PAGE 23 — Trapezoids take two slope facts

11.4 Exactly One Pair

FIGURE: fig11-trapezoids-in-the-plane.png (full width)

Left figure — the two slope facts:

$m_{AB} =$ ____ and $m_{DC} =$ ____ , so the bases ____________________

$m_{AD} =$ ____ and $m_{BC} =$ ____ , so the legs ____________________
Why can the second fact never be skipped?
Right figure — leg lengths ______ and ______ , so the family is ____________________
Right figure — $AC =$ ______ , $BD =$ ______ , midpoints ______ and ______

Together these show ____________________

Congruent diagonals do not make a rectangle. They make a rectangle in a parallelogram. Check the parallelogram first.

PAGE 24 — Choosing the proof

One Question at a Time, Cheapest First

FIGURE: fig12-choosing-the-efficient-proof.png (full width)

Which question is asked first, and why that one?
Count the computations: square ______ isosceles trapezoid ______

A correct long proof still earns the credit. The point of choosing well is that there is less to get wrong.

PAGE 25 — Practice · name the family

Most Specific Family, With Proof

$A(-4,-3)$ , $B(4,-3)$ , $C(3,3)$ , $D(-1,3)$ . ______________
$A(-4,-3)$ , $B(4,-3)$ , $C(2,3)$ , $D(-2,3)$ . ______________
$A(-5,-2)$ , $B(5,-2)$ , $C(2,4)$ , $D(-3,4)$ . ______________
$A(-5,-2)$ , $B(5,-2)$ , $C(3,4)$ , $D(-3,4)$ . ______________

PAGE 26 — Practice · the individual checks

Practice

For $A(-4,-3)$ , $B(4,-3)$ , $C(2,3)$ , $D(-2,3)$ unless told otherwise.

The four slopes: ____ ____ ____ ____ The two facts they establish: ____________________
$AD$ and $BC$ : ______ ______ Family: ____________________
$AC$ and $BD$ : ______ ______
Midpoints of $\overline{AC}$ and $\overline{BD}$ : ______ ______ Their being different proves ____________________
$AD$ and $BC$ for $A(-4,-3)$ , $B(4,-3)$ , $C(3,3)$ , $D(-1,3)$ : ______ ______
One pair of parallel sides and one pair that is not. Family: ______________
Both pairs parallel. Why can it not be a trapezoid in this course?

PAGE 27 — Think it through · 11.4

Think It Through

Application. A roof truss panel has corners $A(-4,-3)$ , $B(4,-3)$ , $C(2,3)$ , $D(-2,3)$ , in feet. Prove it is an isosceles trapezoid, and give each leg exactly and to the nearest hundredth.
Application. In as few computations as possible, decide whether $A(-5,1)$ , $B(-1,-2)$ , $C(5,2)$ , $D(1,5)$ is a rectangle. Say what you compute, in what order, and answer.
Error analysis. A student shows $m_{AB} = m_{DC}$ and writes "so $ABCD$ is a trapezoid." What is missing, and why does it matter?
Error analysis. A student finds a figure's diagonals are congruent and perpendicular and concludes "square." Using item 94's figure, show the conclusion can be false, and name the check that was skipped.
Reasoning. Why is the midpoint question worth asking before any other, whatever family you suspect?

Exit ticket 11.4

The two slope facts a trapezoid proof needs: ____________________
The extra check that makes it isosceles: ____________________
Is $A(-4,-3)$ , $B(4,-3)$ , $C(2,3)$ , $D(-2,3)$ a parallelogram? ______ Deciding computation: ____________
First question to ask about any quadrilateral: ____________________ Formula: ____________

PAGE 28 — Blank grids

Your Turn

FIGURE: fig14-blank-coordinate-frames.png (full width)

A checklist for every problem in this chapter:

Plot the four points. In order — $A$ to $B$ to $C$ to $D$ and back.
Which claim does the question need — parallel, perpendicular, congruent, or bisects?
That names the formula. Write the formula down before computing.
Is it a parallelogram? (Two midpoints. Ask this first, always.)
Finish with the line that names the property.

PAGE 29 — Chapter review

Chapter 11 Review

Review 1 (G.PC.1b). For $A(-3,-1)$ , $B(3,-3)$ , $C(5,3)$ , $D(-1,5)$ :

Show it is a parallelogram in two computations, and name the property used.
Show it is a rhombus in two more, and name the property used.
Show it is a rectangle in two more, and name the property used.
Name the most specific family, and give the total. Then say how many it would have taken to compute all four sides, all four slopes, and both diagonals.

PAGE 30 — Chapter review, continued

Review 2 (G.PC.1 a, b). Name the most specific family and the one computation that decided it.

Points	Family	The deciding computation
$A(0,0)$ , $B(6,2)$ , $C(7,6)$ , $D(1,4)$
$A(-6,0)$ , $B(0,-3)$ , $C(6,0)$ , $D(0,3)$
$A(-4,1)$ , $B(0,-3)$ , $C(6,3)$ , $D(2,7)$
$A(-4,-3)$ , $B(4,-3)$ , $C(3,3)$ , $D(-1,3)$
$A(-4,-3)$ , $B(4,-3)$ , $C(2,3)$ , $D(-2,3)$

Review 3 (G.PC.1b). A designer places three corners of a parallelogram panel at $A(-6,-2)$ , $B(0,-4)$ , $C(4,2)$ , in centimetres.

Find $D$ , showing the algebra rather than the picture.
Give both side lengths exactly, and say whether the panel is a rhombus.
Give both diagonal lengths exactly, and say whether the panel is a rectangle.
The designer now wants a square panel with those same three corners. Explain why no choice of $D$ can do it, using a computation you have already made.

PAGE 31 — Vocabulary check

Words to Know

coordinate plane · slope formula · distance formula · midpoint formula · parallel · perpendicular · congruent · bisect · simplest radical form · coordinate proof · converse · most specific family

The three that carry the chapter:

Slope → parallel or perpendicular. The only angle it can certify is $90°$ , and it cannot certify even that when a segment is vertical.
Distance → congruent. Exact radicals only. Two decimals agreeing is not a proof.
Midpoint → bisects. Two computations decide parallelogram or not, which is the question everything else depends on.

Answer keys for every item are in Appendix A.