Right, acute or obtuse?

Pythagoras works backwards: square the two shorter sides and add, then compare with the longest side squared. Equal: right-angled. The longest square is bigger: obtuse. Smaller: acute.

Given only three lengths, can you tell whether a triangle has a right angle, without drawing it? Builders do exactly this when they mark out a square corner with a tape measure.

Pythagoras' theorem only works for right-angled triangles, and that is what makes it a test. Take the three sides of any triangle, call the longest one $c$ , and compare:

If…	the angle opposite $c$ is…	the triangle is…
$a^2 + b^2 = c^2$	90°	right-angled
$a^2 + b^2 < c^2$	more than 90°	obtuse
$a^2 + b^2 > c^2$	less than 90°	acute

Whole numbers that fit exactly, like 3, 4, 5, are called Pythagorean triples.

Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.

PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.

4, 5 and 6

Compare $4^2 + 5^2$ with $6^2$ .

A triangle has sides 4 cm, 5 cm and 6 cm. Is it right-angled, acute or obtuse?AcuteObtuseRight-angled

TryTry it on the board and compare the result with your prediction.

When do the squares balance?

Open the angle and Close the angle change the corner between a and b.

Try it yourselfYour edits are saved in this browser.

The angle between a and b is 70°. Change it until the two smaller squares exactly make the biggest. What angle is that? Then compare with your prediction.

Use the board. This ticks when you get there.

InvestigateWork out why it happens, one step at a time.

Compare the squares

Here is the test, step by step.

Worked exampleBefore each step, say what you think comes next.

$c = 13 \text{ (the longest)}$ Choose the longest side as c.
$5^2 + 12^2 = 25 + 144 = 169$ Square the other two and add.
$13^2 = 169$ Square the longest.
$169 = 169 \Rightarrow \text{right-angled}$ Equal: right-angled. Less: obtuse. More: acute.

Now try one: sides 6, 8 and 11.

Read Ava's working. Which side must be c?

A triangle has sides 6 cm, 8 cm and 11 cm. Is it right-angled, acute or obtuse?Right-angledAcuteObtuse

Ava tests whether a triangle is right-angled. Click the line where her method goes wrong.

3, 4, 5 is a Pythagorean triple. Which of these is also one?3, 4, 66, 8, 104, 5, 65, 6, 7

ModifyChange the problem on purpose and see what happens.

One change at a time

Each task changes one thing.

Need a hint?

Close the angle. Watch c² fall below 25.

Try it yourselfYour edits are saved in this browser.

Close the angle until c² is less than 20.

Use the board. This ticks when you get there.

Is a triangle with sides 7, 24 and 25 right-angled?Yes: 49 + 576 = 625 = 25²No: 24² + 25² is not 7²No: 7 + 24 is not 25

Two sides of a triangle are 5 and 12. What is the smallest whole-number longest side that makes it obtuse?

Edge case: what kind of triangle has sides 2 cm, 3 cm and 6 cm?ObtuseAcuteNone: 2 + 3 is less than 6, so the sides cannot meet

PractiseA few on your own, to check you have it.

Right, acute or obtuse

Sides 9, 12 and 15?ObtuseRight-angledAcute

Sides 5, 6 and 7?ObtuseRight-angledAcute

Sides 3, 5 and 7?Right-angledObtuseAcute

Review: hypotenuse 10 cm, one side 6 cm. The other side, in cm?

MakeMake your own problem using what you have learned. Expect a few tries; that is normal.

A new triple

Then: is the triple you found a multiple of another one?

Try it yourselfYour edits are saved in this browser.

Need a hint?

Try a = 5 and look along the b values.

Another hint

Or a = 8: which b makes a² + b² a perfect square?

Find a right-angled triangle with whole-number sides, hypotenuse under 20, that is not 3-4-5 or a multiple of it.

Use the board. This ticks when you get there.

For teachers

Students open and close the angle between two sides to see that c² = a² + b² only at 90°, then use the comparison to classify triangles from their side lengths as right-angled, acute or obtuse, and hunt for Pythagorean triples (AC9M8M06, node 3). About 40 minutes.

Open the teacher guide: answers, solutions, slides and a worksheet.

Students will be able to

I can tell from three side lengths whether a triangle is right-angled, acute or obtuse.
I can explain why the longest side must be c in the test.
I can find and check Pythagorean triples.

Before class

Do the Run board yourself first. Students need Pythagoras (previous two lessons) and acute, right and obtuse triangles (Classifying triangles).

The one idea: compare a² + b² with the longest side squared; equal means right-angled, and the bigger angle opposite c pushes c² up. The common mistake is testing with a shorter side as c (Ava's mistake).

The test is the converse of Pythagoras' theorem: if a² + b² = c², the angle opposite c is a right angle. It is a separate fact from the theorem (which starts from a right angle); the board's angle slider shows why it holds.

The board's angle slider keeps a and b fixed and works out c, so students see c² change continuously.

Running the lesson

Predict: take a vote; most guess right-angled because the numbers are consecutive.
Run: ask what happens to c as the angle opens, and to c².
Investigate: build the three-row table on the board from the Run observations.
Modify: for 2, 3, 6, ask students to try to build it with straws or strips.
Make: list the triples the class finds and spot multiples.

The hard part (on purpose)

Remembering which way the inequality goes. Ask students to open the angle on the board and say what happened to c² before they use the rule, instead of giving them the table.

When students get stuck

A student uses a shorter side as c (Ava's mistake). Ask: Which side is opposite the biggest angle? Fix: c is always the longest side.
A student mixes up acute and obtuse. Ask: If the angle opens wider, does the opposite side get longer or shorter? Fix: Wider angle, longer c, bigger c²: obtuse.
A student compares the sides, not the squares. Ask: What does the theorem say: sides or squares? Fix: Square all three first.

Stretch and support

Stretch: Euclid's recipe: for whole numbers m > n, the numbers m² − n², 2mn and m² + n² make a triple. Test it for m = 2, n = 1 and m = 3, n = 2.

Support: Use the decision table and a list of squares to 20²; check with the board.

Where it leads

This completes AC9M8M06.

Where it leads: the converse of Pythagoras in proofs, the cosine rule (Year 11), and distances on the Cartesian plane.

What each step is for

4, 5 and 6: Committing to a type for sides 4, 5, 6 shows whether they compare squares or guess from the shape of the numbers.
when do the squares balance?: Opening and closing the angle between sides 3 and 4 shows that c² equals a² + b² only at 90°, and is bigger beyond it and smaller before it.
compare the squares: Compare a² + b² with the longest side squared; Ava uses a shorter side as c. Multiples of a triple are triples too.
one change at a time: One change at a time: an acute triangle on the board, a triple to test, the smallest obtuse side, then the edge case of lengths that make no triangle at all.
right, acute or obtuse: Each type once and a review, mixed; testing with the wrong side as c is the planted slip.
a new triple: Searching for a Pythagorean triple that is not 3-4-5 or a multiple of it turns the test into a hunt for two squares that add to a square.

Now you can

Tell from three side lengths whether a triangle is right-angled, acute or obtuse.
Explain why the longest side must be c in the test.
Find and check Pythagorean triples.

Practise this skillNew questions, new numbers each timeGet 3 right in a row to mark it practised.Next lesson12. Composite shapesA composite shape is made of shapes you know. Cut it into pieces and add their areas, or see it as a big rectangle with pieces taken away; the area is the same either way. The perimeter is the distance round the outside.

Source: content/maths/measurement/right-or-not/lesson.md