Algorithms for classifying shapes

A triangle with three equal sides is rotated on the page. Should an algorithm based on side equality give a different most specific name?

Specify inputs, decisions and outputs

An algorithm is a sequence of unambiguous instructions. A decision selects a branch using a stated condition

First specify the allowed inputs and the required output. Then choose the properties needed to distinguish those outputs

A Yes branch is followed when its condition is true; a No branch is followed when it is false

A terminal output ends that run. A decision after that stopping point is not reached, which is different from testing it and getting No

We use 1 for a Yes trace, 0 for a No trace and 9 for not reached. Terminal name codes have their own legend in each question

State the domain before designing branches

The triangle classifiers here receive valid triangles with consistent side and angle facts

If inputs were just three proposed lengths, first check that they are positive and that the two shorter lengths add to more than the longest

Our polygon domain has a closed straight-sided boundary without crossings and no straight interior angles

These assumptions matter: a classification rule for valid simple polygons should not silently accept an open or self-crossing drawing

Test the narrowest side class first

Isosceles means at least two equal sides, so equilateral triangles also satisfy that broader definition

For one most specific name, test all three equal before any equal pair

On the No branch from A, an equal pair means exactly two equal sides. The branch history supplies that restriction

Construct the two decision slots

Complete this fixed triangle-side skeleton. The first Yes branch must output equilateral; its No branch leads to a second test whose Yes branch outputs isosceles and No branch outputs scalene

Test bank: 1 = all three sides equal; 2 = any pair of sides equal; 3 = all angles acute

Enter the test code for each slot. Keep the stated branch order

Construct the specified algorithm by filling each missing slot with its bank code

Use exact whole-number codes from the question, with no units

Equilateral is the most specific name when all three sides are equal

Put test 1 first so an equilateral triangle reaches its intended output

After a No, the triangle does not have three equal sides

Test 2 now distinguishes exactly two equal sides from no equal pair

The completed test sequence is 1, then 2

Stop before the second test

Inputs are valid triangles. Output the most specific side-length name

Decision A: are all three sides equal? Yes: output 1 (equilateral) and stop. No: go to B

Decision B: is there any equal pair of sides? Yes: output 2 (isosceles) and stop. No: output 3 (scalene) and stop

Trace codes: 1 means Yes, 0 means No, and 9 means the decision was not reached. Output codes are given in the algorithm

Triangle ABC has these side relationships: AB = BC = CA

Its interior angles are A = 60°, B = 60° and C = 60°

Trace the supplied algorithm and complete every requested entry

Use exact whole-number codes from the question, with no units

Read the supplied properties and start at decision A

Read the side relationships rather than the orientation

AB = BC = CA

All three sides are equal, so equilateral is the most specific name

Decision A gives Yes

Follow the Yes branch to its terminal output. Decision B is not reached, so record 9

Record terminal output 1: equilateral

Not reached is different from a tested No result

Follow a No branch before a Yes branch

Reach the final side branch

Repair an early stopping error

Draft algorithm for the most specific side name: if any pair of sides is equal, output 2 (isosceles) and stop; otherwise output 3 (scalene)

Correct output codes: equilateral = 1, isosceles = 2, scalene = 3

Repair bank: 1 = test all three equal first and output equilateral on Yes, then use the draft on No; 2 = rotate the drawing; 3 = keep the draft unchanged

Triangle ABC has these side relationships: AB = BC = CA

Its interior angles are A = 60°, B = 60° and C = 60°

Trace the faulty algorithm, record the required result and enter the repair code

Use exact whole-number codes from the question, with no units

The given equilateral triangle has equal pairs, so the first draft condition is true

The draft stops with code 2, isosceles

That broader name applies, but the specification requests the most specific name: equilateral, code 1

Testing all three equal before any equal pair fixes this counterexample

Choose repair 1. Rotation does not alter the equality facts

Classify using the largest angle

An acute triangle needs all three angles below 90°. One acute angle is not enough

Checking the largest angle handles all three cases with two decisions

After both No branches, the largest angle is below 90°, so every angle is acute

Match tests to their required branch outputs

Complete this fixed triangle-angle skeleton. The first Yes branch must output obtuse; its No branch leads to a second test whose Yes branch outputs right-angled and No branch outputs acute

Test bank: 1 = largest angle equals 90°; 2 = largest angle exceeds 90°; 3 = smallest angle is less than 90°

Enter the test code for each slot. Keep the stated branch order

Construct the specified algorithm by filling each missing slot with its bank code

Use exact whole-number codes from the question, with no units

An obtuse triangle is identified by its largest angle exceeding 90°

Put test 2 first

After a No, the largest angle is at most 90°

Put test 1 next to separate equality from an acute largest angle

The completed test sequence is 2, then 1

Record a skipped second angle test

Inputs are valid triangles. Compare the largest interior angle with 90°

Decision A: is the largest angle greater than 90°? Yes: output 3 (obtuse) and stop. No: go to B

Decision B: is the largest angle equal to 90°? Yes: output 2 (right-angled) and stop. No: output 1 (acute) and stop

Trace codes: 1 means Yes, 0 means No, and 9 means the decision was not reached. Output codes are given in the algorithm

Triangle ABC has these side relationships: CA = CB, and AB has a different length

Its interior angles are A = 30°, B = 30° and C = 120°

Trace the supplied algorithm and complete every requested entry

Use exact whole-number codes from the question, with no units

Read the supplied properties and start at decision A

Compare all three angles with 90°

The largest angle is 120°

One angle is greater than 90°, so the triangle is obtuse

Decision A gives Yes

Follow the Yes branch to its terminal output. Decision B is not reached, so record 9

Record terminal output 3: obtuse

Not reached is different from a tested No result

Handle equality at ninety degrees

Use two No branches

Test a classifier that checks only one acute angle

Add names rather than choosing only one

A different specification can ask for every applicable name, not one most specific name

Both opposite-side pairs parallel gives parallelogram. Within that class, four right angles gives rectangle and four equal sides gives rhombus

A square satisfies both conditions, so it belongs to all four classes: parallelogram, rectangle, rhombus and square

Use independent additions for these overlapping classes. Do not stop after adding rectangle if rhombus and square tests still apply

Make the rest of the scheme explicit

This scheme uses simple convex quadrilaterals. Trapezium means exactly one pair of opposite sides is parallel

For this scheme, kite means two pairs of adjacent equal sides, with different lengths for the two pairs. Rhombuses and squares therefore do not enter this particular kite output

Terminology can use other inclusion conventions. A classifier must declare its definitions rather than change them between inputs

If there are two parallel pairs, use the parallelogram tests. Otherwise, check exactly one pair for trapezium; with no parallel pairs, test the stated kite condition

A shape satisfying none of these conditions receives none of these six names. The empty output list is still a valid result for that domain

Change the width through the five whole-number settings. The height stays three units and every angle stays a right angle. Read 1 as included and 0 as not included in the output list

What should the algorithm retain when width changes from two to three units?

Construct independent tests for overlapping classes

A valid quadrilateral has already passed the two-parallel-pair test, so parallelogram is in its output list

Complete three independent tests in this order: add rectangle when slot A is true; add rhombus when slot B is true; add square when slot C is true

Test bank: 1 = four right angles; 2 = four equal sides; 3 = both conditions 1 AND 2; 4 = either condition 1 OR 2

Construct the specified algorithm by filling each missing slot with its bank code

Use exact whole-number codes from the question, with no units

Rectangle requires four right angles, so A uses test 1

Rhombus requires four equal sides, so B uses test 2

Square requires both conditions together, so C uses test 3

These are independent tests, not an else-if chain

A square therefore keeps parallelogram, rectangle, rhombus and square

Keep only the justified names

Inputs are simple convex quadrilaterals. Clear the output list for each new shape

In this scheme, trapezium means exactly one parallel pair. Kite means two pairs of adjacent equal sides with different lengths for the two pairs

If both pairs of opposite sides are parallel, add parallelogram. Then independently test: four right angles adds rectangle; four equal sides adds rhombus; both conditions together adds square

Otherwise, if exactly one pair of opposite sides is parallel, add trapezium. Otherwise, test the stated kite condition and add kite when it holds

Keep every name added to the list. An empty list means none of these six names applies

For each named output, enter 1 if the algorithm adds it and 0 if it does not

Quadrilateral ABCD has the following stated properties

AB is parallel to CD and BC is parallel to DA. AB = CD = 6 cm and BC = DA = 4 cm. Angle A is 60°

Trace the supplied algorithm and complete every requested entry

Use exact whole-number codes from the question, with no units

Start with an empty output list

Read the stated side and angle properties

AB is parallel to CD and BC is parallel to DA. AB = CD = 6 cm and BC = DA = 4 cm. Angle A is 60°

Both pairs of opposite sides are parallel, so parallelogram also applies

Select every matching name: parallelogram

After every applicable test, the complete list is parallelogram

parallelogram: included, so enter 1

rectangle: not included, so enter 0

rhombus: not included, so enter 0

square: not included, so enter 0

kite: not included, so enter 0

trapezium: not included, so enter 0

Do not erase a broader name when a more specific name is added

Keep a broader class when adding rectangle

Separate equal sides from right angles

Retain four names for a square

Inputs are simple convex quadrilaterals. Clear the output list for each new shape

In this scheme, trapezium means exactly one parallel pair. Kite means two pairs of adjacent equal sides with different lengths for the two pairs

If both pairs of opposite sides are parallel, add parallelogram. Then independently test: four right angles adds rectangle; four equal sides adds rhombus; both conditions together adds square

Otherwise, if exactly one pair of opposite sides is parallel, add trapezium. Otherwise, test the stated kite condition and add kite when it holds

Keep every name added to the list. An empty list means none of these six names applies

For each named output, enter 1 if the algorithm adds it and 0 if it does not

Quadrilateral ABCD has the following stated properties

All four sides are equal and all four angles are right angles

Trace the supplied algorithm and complete every requested entry

Use exact whole-number codes from the question, with no units

Start with an empty output list

Read the stated side and angle properties

All four sides are equal and all four angles are right angles

Equal sides and four right angles make a square

Four right angles make a rectangle

Four equal sides make a rhombus

Both pairs of opposite sides are parallel, so parallelogram also applies

Select every matching name: square, rectangle, rhombus, parallelogram

After every applicable test, the complete list is square, rectangle, rhombus, parallelogram

parallelogram: included, so enter 1

rectangle: included, so enter 1

rhombus: included, so enter 1

square: included, so enter 1

kite: not included, so enter 0

trapezium: not included, so enter 0

Do not erase a broader name when a more specific name is added

Repair an exclusive branch in an inclusive hierarchy

Follow the non-parallel branch

Inputs are simple convex quadrilaterals. Clear the output list for each new shape

In this scheme, trapezium means exactly one parallel pair. Kite means two pairs of adjacent equal sides with different lengths for the two pairs

If both pairs of opposite sides are parallel, add parallelogram. Then independently test: four right angles adds rectangle; four equal sides adds rhombus; both conditions together adds square

Otherwise, if exactly one pair of opposite sides is parallel, add trapezium. Otherwise, test the stated kite condition and add kite when it holds

Keep every name added to the list. An empty list means none of these six names applies

For each named output, enter 1 if the algorithm adds it and 0 if it does not

Quadrilateral ABCD has the following stated properties

AB = AD and BC = CD, with the first pair longer than the second pair. No opposite sides are parallel

Trace the supplied algorithm and complete every requested entry

Use exact whole-number codes from the question, with no units

Start with an empty output list

Read the stated side and angle properties

AB = AD and BC = CD, with the first pair longer than the second pair. No opposite sides are parallel

Two pairs of adjacent equal sides give a kite

Select every matching name: kite

After every applicable test, the complete list is kite

parallelogram: not included, so enter 0

rectangle: not included, so enter 0

rhombus: not included, so enter 0

square: not included, so enter 0

kite: included, so enter 1

trapezium: not included, so enter 0

Do not erase a broader name when a more specific name is added

Stop at exactly one parallel pair

Allow an empty list when no named class applies

Separate side count from regularity

First count the sides. Then test the side-length and angle conditions independently

Regular means all side lengths equal AND all interior angles equal. If either condition fails, the polygon is irregular

OR is different: it accepts a shape when just one condition holds, which would incorrectly accept a non-square rectangle or non-square rhombus

Keep the side-count output even when the polygon is irregular. An irregular five-sided polygon is still a pentagon

Test for a reflex interior angle

In the stated simple-polygon domain, an interior angle greater than 180° gives a concave polygon

If no interior angle is reflex and straight angles have been excluded, the polygon is convex

The inward corner still contributes one vertex and two meeting boundary sides

Convexity and regularity answer different questions. A non-square rectangle is convex and irregular; the notched pentagon is concave and irregular

Construct the two-condition decision

Complete: if all side lengths are equal [operator] all interior angles are equal, output [Yes code]; otherwise output [No code]

Operator bank: 1 = AND; 2 = OR. Output bank: 1 = regular; 2 = irregular

The algorithm must correctly classify every simple polygon in its domain

Construct the specified algorithm by filling each missing slot with its bank code

Use exact whole-number codes from the question, with no units

Regularity requires the side condition and the angle condition together

Use AND, operator code 1

When both hold, output regular, code 1

If either condition fails, output irregular, code 2

A non-square rectangle or non-square rhombus disproves the OR version

Run the complete polygon scheme on a pentagon

Inputs are simple polygons: straight sides form a closed boundary without crossing, and no interior angle is exactly 180°

Count every side once around the boundary, including the sides meeting at an inward vertex

If all sides are equal AND all interior angles are equal, output regular; otherwise output irregular

If any interior angle is greater than 180°, output concave; otherwise output convex

Output codes: regular = 1, irregular = 2; convex = 1, concave = 2

This closed polygon has 5 straight sides

All its side lengths are equal and all its interior angles are equal

Trace the supplied algorithm and complete every requested entry

Use exact whole-number codes from the question, with no units

Follow the boundary before testing its properties

Count the 5 sides once around the closed boundary

There are also 5 vertices

A 5-sided polygon is a pentagon

Check both conditions for regularity

Every side length is equal and every interior angle is equal

Both conditions hold, so the polygon is regular

Side-count output: 5

Both equal-side and equal-angle conditions hold, so the regularity code is 1

Every interior angle is less than 180°, so the convexity code is 1

An irregular polygon can be either convex or concave

Record side count and separate property outputs

Reject equal angles alone as sufficient

Inputs are simple polygons: straight sides form a closed boundary without crossing, and no interior angle is exactly 180°

Count every side once around the boundary, including the sides meeting at an inward vertex

If all sides are equal AND all interior angles are equal, output regular; otherwise output irregular

If any interior angle is greater than 180°, output concave; otherwise output convex

Output codes: regular = 1, irregular = 2; convex = 1, concave = 2

All four angles are right angles. AB = CD = 6 cm and BC = DA = 4 cm

Trace the supplied algorithm and complete every requested entry

Use exact whole-number codes from the question, with no units

Follow the boundary before testing its properties

Trace the closed boundary

There are four straight sides

A four-sided polygon is a quadrilateral

A regular polygon must have both equal sides and equal angles

The side lengths include both 6 cm and 4 cm

One of the two necessary conditions fails, so this polygon is irregular

Side-count output: 4

Both equal-side and equal-angle conditions do not hold together, so the regularity code is 2

Every interior angle is less than 180°, so the convexity code is 1

An irregular polygon can be either convex or concave

Reject equal sides alone as sufficient

Use a rectangle as a counterexample to OR

Construct the convexity branch

Keep the inward vertex in the trace

Inputs are simple polygons: straight sides form a closed boundary without crossing, and no interior angle is exactly 180°

Count every side once around the boundary, including the sides meeting at an inward vertex

If all sides are equal AND all interior angles are equal, output regular; otherwise output irregular

If any interior angle is greater than 180°, output concave; otherwise output convex

Output codes: regular = 1, irregular = 2; convex = 1, concave = 2

The five straight boundary segments join in order A to B to C to D to E to A without crossing

Vertex D points into the shape and its interior angle is reflex. The side lengths are not all equal

Trace the supplied algorithm and complete every requested entry

Use exact whole-number codes from the question, with no units

Follow the boundary before testing its properties

Count AB, BC, CD, DE and EA

The inward corner does not remove either of its two sides

Five sides make a pentagon

The closed boundary has straight sides and no crossings, so it is a polygon

Its unequal side lengths make it irregular

The reflex interior angle at D makes it concave

Side-count output: 5

Both equal-side and equal-angle conditions do not hold together, so the regularity code is 2

There is a reflex interior angle, so the convexity code is 2

An irregular polygon can be either convex or concave

Repair a counter that skips inward vertices

Repeat for the whole set

For each input shape, clear its output record, run the specified tests and record the complete result

Then advance to the next shape and repeat. Stop when no input remains

Clearing prevents the previous shape’s names from being copied into the next result

A triangle can receive a side classification and an angle classification at the same time. Run both algorithms rather than treating the two labels as competing answers

Assemble a complete per-shape process

Classify three inputs by two properties

Inputs are valid triangles. Output the most specific side-length name

Decision A: are all three sides equal? Yes: output 1 (equilateral) and stop. No: go to B

Decision B: is there any equal pair of sides? Yes: output 2 (isosceles) and stop. No: output 3 (scalene) and stop

Inputs are valid triangles. Compare the largest interior angle with 90°

Decision A: is the largest angle greater than 90°? Yes: output 3 (obtuse) and stop. No: go to B

Decision B: is the largest angle equal to 90°? Yes: output 2 (right-angled) and stop. No: output 1 (acute) and stop

Process shapes 1, 2 and 3 in their supplied order. Start both algorithms afresh for each shape and record its two terminal codes

Shape 1

Triangle ABC has these side relationships: AB = BC = CA

Its interior angles are A = 60°, B = 60° and C = 60°

Shape 2

Triangle ABC has these side relationships: CA = CB, and AB has a different length

Its interior angles are A = 70°, B = 70° and C = 40°

Shape 3

Triangle ABC has these side relationships: AB, BC and CA all have different lengths

Its interior angles are A = 30°, B = 60° and C = 90°

Run both algorithms for every supplied shape and enter the paired terminal codes

Use exact whole-number codes from the question, with no units

Take each shape separately; do not carry a previous output to the next shape

Shape 1: Triangle ABC has these side relationships: AB = BC = CA; Its interior angles are A = 60°, B = 60° and C = 60°

Side decisions give equilateral, code 1. Angle decisions give acute, code 1

Shape 2: Triangle ABC has these side relationships: CA = CB, and AB has a different length; Its interior angles are A = 70°, B = 70° and C = 40°

Side decisions give isosceles, code 2. Angle decisions give acute, code 1

Shape 3: Triangle ABC has these side relationships: AB, BC and CA all have different lengths; Its interior angles are A = 30°, B = 60° and C = 90°

Side decisions give scalene, code 3. Angle decisions give right, code 2

Record both classifications for each input, then stop after the third input

Produce both codes for every input

Find names left over from the previous shape

Inputs are simple convex quadrilaterals. Clear the output list for each new shape

In this scheme, trapezium means exactly one parallel pair. Kite means two pairs of adjacent equal sides with different lengths for the two pairs

If both pairs of opposite sides are parallel, add parallelogram. Then independently test: four right angles adds rectangle; four equal sides adds rhombus; both conditions together adds square

Otherwise, if exactly one pair of opposite sides is parallel, add trapezium. Otherwise, test the stated kite condition and add kite when it holds

Keep every name added to the list. An empty list means none of these six names applies

Faulty list processing: classify shape 1, then classify shape 2 without clearing the output list; new names are added but previous names remain

Shape 1 is the first supplied figure and shape 2 is the second. For the requested names use 1 = present, 0 = absent

Repair bank: 1 = clear the output list before classifying each shape; 2 = retain the earlier names; 3 = stop after shape 1

Shape 1

Quadrilateral ABCD has the following stated properties

All four sides are equal and all four angles are right angles

Shape 2

Quadrilateral ABCD has the following stated properties

AB is parallel to CD. AD and BC are not parallel. The four side lengths are different

Trace the faulty algorithm, record the required result and enter the repair code

Use exact whole-number codes from the question, with no units

Shape 1 is a square and leaves parallelogram, rectangle, rhombus and square in the list

Shape 2 has exactly one parallel pair and should receive only trapezium under this scheme

The faulty process adds trapezium but retains rectangle from shape 1

Thus its rectangle entry is 1 instead of 0, while its trapezium entry is correctly 1

Repair 1 clears the list before each new input and then records its own complete classification

Compare expected and actual outputs

Write the expected result from the geometric definitions before running a draft algorithm

Choose inputs that reach different branches, including equality boundaries and overlapping classes

For triangle angles, include acute, right and obtuse inputs. For the hierarchy, a square tests whether multiple applicable names survive

A counterexample is one allowed input whose actual output differs from the required output. One counterexample disproves correctness

Passing a finite set of tests is useful evidence but not a proof for every input. Explain why the conditions and branches cover the full stated domain

A triangle-angle classifier checks the largest angle. Which test plan best probes its three outputs and the equality boundary?

Design and explain a classification scheme

Choose a set containing an equilateral triangle, a right-angled triangle, a square, a non-square rectangle and a concave pentagon

State the domain and required output names, then draw a flowchart or write a decision table that first separates shapes by side count and applies the relevant property tests

Specify whether each branch returns one most specific name or every applicable name. Make the kite and trapezium conventions explicit if those names are included

Trace every supplied shape, record expected and actual outputs, and refine any branch that loses a required class

Explain why each decision is needed and why all allowed inputs have a result. This written design and explanation are not automatically checked

Make each branch serve the specification

State the domain, allowed properties and required outputs before selecting decisions

Use ordered branches for one most specific name, but independent additions when all overlapping names are required

Keep No and not reached distinct in a trace

Regularity needs both equal sides and equal angles; concavity uses a reflex interior angle

Reset the output record for each new shape and stop after processing the whole set

Use counterexamples to repair errors, then justify the complete decision structure rather than treating a few passing tests as proof