Notebook

00:06

12 Apr 2026

The Four-Shape Decision Tree

Welcome! Today we're tackling The Four-Shape Decision Tree — the systematic method that turns your distance calculations into a definitive answer.

You can now compute all six distances for any quadrilateral.

But what do the numbers mean?

Here's the problem:

What you observe	Could be...
All sides equal	Square OR Rhombuseither!
Opposite sides equal	Rectangle OR Parallelogrameither!

The numbers alone do not tell you — you need a systematic decision process.

In this section, you will learn the decision tree that converts six distance values into a definitive classification.

Key insight: The diagonal comparison is always the tiebreaker.

1. The side-equality checks: all equal vs opposite equal

We start with the first level of the decision tree: checking the sides.

There are two possible outcomes:

All four sides are equal
Only opposite sides are equal

This first check narrows the classification from four shapes down to two.

📋 Given Info

After computing all four side-lengths, you compare them. There are two important outcomes:

All four sides are equal: $A B = B C = C D = D A$
- Shape is either a square or a rhombus
Opposite sides are equal but not all four: $A B = C D$ and $B C = D A$ , but $A B \neq B C$
- Shape is either a rectangle or a parallelogram

This single comparison splits the four shapes into two groups of two.

✍️ Question

Your Turn 🤔

Two quadrilaterals have been computed:

Quadrilateral 1: $A B = 6, B C = 6, C D = 6, D A = 6$

Quadrilateral 2: $A B = 8, B C = 5, C D = 8, D A = 5$

For each quadrilateral:

What does the side comparison tell you?
Which two shapes are still possible?

The decision tree starts with a side comparisonAlways check sides first:

Check 1: Are all four sides equal?Check this before anything else

If $AB = BC = CD = DA$ Equal sides alone doesn't tell you which shape (all the same number), the shape has equal sides.

But this does NOT automatically mean squareThis is a common mistake to avoid. A rhombus also has all four sides equal.

Side-by-side comparison: a square ABCD with all sides labeled 's' and a rhombus ABCD with all sides labeled 's', showing both have equal sides but different angles

So 'all sides equal' narrows it to: SQUARE or RHOMBUSNeed more information to decide which one.

✍️ MCQ

Choose one

Quadrilateral 1 has

AB = 6

BC = 6

CD = 6

DA = 6

. Based on the side comparison alone, which two shapes are still possible?

✓

Let's annotate our quadrilateral to highlight the key question: when all sides are equal, how do we decide between a square and a rhombus?

Check 2: If not all equal, are opposite sides equal?

If $A B = C D$ and $B C = D A$ (opposite pairs match, but the two pairs are different numbers), the shape has the parallelogram propertyOpposite sides come in matching pairs, but pairs differ in length.

Parallelogram ABCD with sides labeled: AB=CD=5cm, BC=DA=3cm, showing opposite sides equal but not all sides equal

But this does NOTBoth rectangle and parallelogram share this property automatically mean rectangle.

So 'opposite sides equal' narrows it to: RECTANGLE or PARALLELOGRAMYou need another check to tell them apart.

✍️ MCQ

Choose one

Quadrilateral 2 has

AB = 8

BC = 5

CD = 8

DA = 5

. Which two shapes could it be?

✓

Now let's apply the same decision tree logic to Quadrilateral 2, where opposite sides are equal but not all four match. This annotation shows the second branch of our decision process.

✍️ MCQ

Choose one

Both quadrilaterals need a tiebreaker. What measurement will distinguish square from rhombus, and rectangle from parallelogram?

✓

In both cases, you cannot make a final decision from sides alone. You MUSTThe only way to make the final decision proceed to the diagonal check.

This is the critical takeaway from our decision tree so far:

Quadrilateral 1 (all sides = 6): Could be square OR rhombus
Quadrilateral 2 (opposite sides equal: 8 and 5): Could be rectangle OR parallelogram

Sides narrow it down to two options, but sides alone neverThey only narrow it down to two possibilities give you the final answer.

✍️ Yes/No

Yes or No?

A quadrilateral has sides

AB = 7

BC = 7

CD = 7

DA = 7

. Can you definitively say it's a square based on this information alone?

✓

⚠️ The common mistake:Don't jump to square just because sides are equal seeing 'all sides = 6' and immediately declaring 'It's a square!'

Look at the board — both shapes have all sides equal to 6Squares and rhombuses both have equal sides. But are they the same shape?

This skips the diagonal checkSkipping diagonals leads to wrong exam answers and will be wrong whenever the shape is actually a rhombus.

Side-by-side comparison: left shows a square with all sides 6 and equal diagonals marked 90 degrees; right shows a rhombus with all sides 6 but diagonals of different lengths, not 90 degrees

Correct approach	Wrong approach
All sides equal → Could be square OR rhombusMust check diagonals to pick between square and rhombus → Must check diagonals	All sides equal → "It's a square!"

Remember: Equal sides narrow it down to two options, not one!Two options remain, not one

✍️ Yes/No

Yes or No?

A student sees a quadrilateral with

AB = BC = CD = DA = 5

and says: "All sides are equal, so it must be a square." Is this reasoning correct?

✓

2. The diagonal tiebreaker: equal vs unequal diagonals

📋 Given Info

The side check narrowed the possibilities to two shapes in each case. Now we bring in the diagonals as the tiebreaker.

Equal diagonals → push toward the 'higher' shape (square or rectangle)
Unequal diagonals → give the 'lower' shape (rhombus or parallelogram)

The Rule:

Side Check Result	Diagonals Equal	Diagonals Unequal
All sides equal	SQUARE	RHOMBUS
Opposite sides equal	RECTANGLE	PARALLELOGRAM

✍️ Question

Your turn! 🔍

Look at the two quadrilaterals with their diagonals now drawn:

Quadrilateral	Side Information	Diagonal $A C$	Diagonal $B D$
Quadrilateral 1	All sides = 6	$\sqrt{40}$	$\sqrt{56}$
Quadrilateral 2	Opposite sides 8 and 5	$\sqrt{89}$	$\sqrt{89}$

Give the final classification for each quadrilateral with justification.

For each one, state: What is it? Why?

The diagonal comparison(The deciding factor between similar shapes) is the tiebreakerWhen sides don't give you the answer. Here is the complete decision tree:

Path 1: All sides equalAsk if diagonals are equal too

Diagonals equal → SQUAREEqual sides plus equal diagonals (all sides equal AND diagonals equal)
Diagonals NOT equal → RHOMBUSEqual sides but different diagonals (all sides equal but diagonals differ)

Side-by-side comparison: square with equal sides and equal diagonals labeled, rhombus with equal sides but unequal diagonals labeled

✍️ MCQ

Choose one

Quadrilateral 1 has all sides equal to

6

, with diagonals

AC = \sqrt{40}

and

BD = \sqrt{56}

. What is it?

✓

Path 2: Opposite sides equalNot all four sides, just opposite pairs (not all four)

Diagonals equal → RECTANGLEOpposite sides equal plus equal diagonals (opposite sides equal AND diagonals equal)
Diagonals NOT equal → PARALLELOGRAMOpposite sides equal but diagonals differ (opposite sides equal but diagonals differ)

✍️ MCQ

Choose one

Quadrilateral 2 has opposite sides

8

and

5

, with diagonals

AC = \sqrt{89}

and

BD = \sqrt{89}

. What is it?

✓

Rhombus with all sides labeled 6, diagonals drawn and labeled sqrt(40) and sqrt(56), showing unequal diagonal lengths

For Quadrilateral 1: sides are all 6First check — are all four sides the same? (Path 1Equal sides means Path 1). Diagonals: $\sqrt{40}$ and $\sqrt{56}$ . Since $40 \neq 56$ , the diagonals are unequalThe tiebreaker — diagonals are different. Path 1 + unequal diagonals = RHOMBUSUnequal diagonals rules out square.

Rectangle with opposite sides labeled 8 and 5, both diagonals drawn and labeled sqrt(89), showing equal diagonal lengths

For Quadrilateral 2: opposite sides are 8 and 5Only opposite pairs match, not all four (Path 2Different from Path 1 — opposite sides equal). Diagonals: both $\sqrt{89}$ . EqualDiagonals decide the final answer. Path 2 + equal diagonals = RECTANGLEEqual diagonals on Path 2 means rectangle.

✍️ MCQ

Choose one

If a quadrilateral has all sides equal to

10

and diagonals of

\sqrt{200}

each, what shape is it?

✓

Now that we've applied the decision tree, we can confirm our classifications. The green checkmarks show the final, verified answers.

The key insight: the diagonal check is never optionalAlways verify diagonals are equal.

'All sides equal' is necessary for a square but not sufficient — you ALSO need equal diagonalsEqual sides alone don't make a square.

Side-by-side comparison: rhombus vs square (both with equal sides, but different diagonals) and parallelogram vs rectangle (both with opposite sides equal, but different diagonals)

'Opposite sides equal' is necessary for a rectangle but not sufficient — you ALSO need equal diagonals.

3. Full classification from raw distance data

You've learned the decision tree in pieces. Now it's time to test the full process.

Given six computed squared distances for a fresh quadrilateral, your task is to run the complete decision tree and arrive at the classification.

The Decision Tree:

Step	Check	Outcome
1	Check sides	All equal → square/rhombus path
		Opposite equal → rectangle/parallelogram path
2	Check diagonals	Equal → square or rectangle
		Unequal → rhombus or parallelogram

✍️ Question

Quadrilateral JKLM has the following squared distances:

Sides	Diagonals
$JK^2 = 20$	$JL^2 = 40$
$K L^{2} = 20$	$K M^{2} = 40$
$L M^{2} = 20$
$M J^{2} = 20$

Run the decision tree and state the classification with full justification.

(Show both checks — don't skip straight to the answer!)

Let's run the decision tree step by step.

Step 1: Check the sides.

$J K^{2} = 20$ , $K L^{2} = 20$ , $L M^{2} = 20$ , $M J^{2} = 20$ .

All four squared sides are 20.We compare the squared values Since they are all equal, all four sides are equalEqual squares mean equal sides: $J K = K L = L M = M J = \sqrt{20} = 2 \sqrt{5}$ .

Quadrilateral JKLM with vertices labeled, all four sides marked equal with tick marks, side length labeled as 2*sqrt(5)

✍️ MCQ

Choose one

Since all four sides of quadrilateral

JKLM

are equal, which path of the decision tree are we on?

✓

This puts us on Path 1Equal sides lead to Path 1 of the decision tree: the shape is either a SQUARESquare or rhombus are the options or a RHOMBUSSquare or rhombus are the options. Diagonals determine the final answer

Now let's annotate the figure to highlight the key question: when all sides are equal, we still don't know if it's a square or rhombus. This is exactly where the decision tree comes in.

✍️ MCQ

Choose one

All sides of

JKLM

are equal (

\sqrt{20}

). What measurement will determine if it's a square or a rhombus?

✓

Step 2: Check the diagonals.This step separates a rhombus from a square

$J L^{2} = 40$ , $K M^{2} = 40$ .

Quadrilateral JKLM with both diagonals JL and KM drawn, labeled with length sqrt(40) or 2sqrt(10), showing they are equal

Both squared diagonals are 40. Since they are equalEqual diagonals plus equal sides means square, the diagonals are equal: $J L = K M = \sqrt{40} = 2 \sqrt{10}$ .

Now we add the final piece of evidence: the diagonals are equal. This confirms the shape is a square, not a rhombus.

Step 3: Combine.

All sides equal (from Step 1) + diagonals equal (from Step 2) = SQUAREThe only shape with yes at both checkpoints.

Look at the decision treeTwo checkpoints — first sides, then diagonals on the board — we followed the "Yes" path at both checkpointsBoth answers yes leads to square!

✍️ MCQ

Choose one

If the diagonals of JKLM had been

JL^2 = 40

and

KM^2 = 32

(unequal), what would the classification be?

✓

If the diagonals had been UNEQUALSame sides but different diagonals changes everything (say $J L^{2} = 40$ but $K M^{2} = 32$ ), the answer would have been RHOMBUSSame sides, different diagonals, completely different answer instead.

The decision tree always has exactly two steps: sides, then diagonalsWhy you must never skip Step 2. Never skip Step 2.The diagonal check separates square from rhombus