Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- A right-angled triangle has legs 6 and 8. How long is the hypotenuse?
Answer
10, because 36 + 64 = 100. - What is the longest side of a right-angled triangle called?
Answer
The hypotenuse. - Which is bigger, 3² + 4² or 6²?
Answer
6² = 36 is bigger than 9 + 16 = 25.
Words used in this lesson
- converse
- the statement turned around: if the squares match, then the angle is 90 degrees. For example: the converse of Pythagoras' theorem.
- Pythagorean triple
- three whole numbers a, b, c with a² + b² = c². For example: 5, 12, 13.
- acute angle
- an angle less than 90 degrees. For example: the corner in a 6, 7, 9 triangle.
- obtuse angle
- an angle between 90 and 180 degrees. For example: the large corner in 7, 8, 11.
- right angle
- an angle of exactly 90 degrees. For example: the corner of a door frame.
A carpenter's rope
A carpenter ties a loop of rope with 12 equal gaps, pins one corner, and stretches the loop so that one side has 3 gaps, the next 4 gaps and the last 5. She says the corner is a perfect right angle without a set square. Before you read on: does the theorem back her up, and how would you check?
The converse: if a triangle's sides satisfy a² + b² = c², where c is the longest side, then the angle opposite c is a right angle.
If a² + b² is more than c² the angle opposite c is acute; if it is less the angle is obtuse. Whole numbers a, b, c with a² + b² = c² are a Pythagorean triple, and any triple multiplied by the same whole number is another triple.
Set three sides and, before the figure answers, say whether the corner is right, acute or obtuse.
36 + 49 = 85 is more than 9² = 81, so the corner opposite 9 is acute.
Show me the steps
- 6, 7, 9: 36 + 49 = 85 and 9² = 81. 85 is more than 81, so the corner is acute.
- 8, 15, 17: 64 + 225 = 289 and 17² = 289. Equal, so it is a right angle.
- 8, 15, 16: 289 against 256. More, so acute. 8, 15, 18: 289 against 324. Less, so obtuse.
- 12, 16, 20: 144 + 256 = 400 and 20² = 400. Equal, a right angle.
- 3, 4, 9: 3 + 4 is less than 9, so the sides cannot meet. There is no triangle.
Why the converse is true
The theorem says “right angle, so a² + b² = c²”. The converse says the opposite way round, and it needs its own argument.
Suppose a triangle has sides a, b and c with a² + b² = c². Show that the angle opposite c is 90°.
- Build a second triangle with a right angle between two sides of lengths a and b.
- By Pythagoras' theorem, its third side is √(a² + b²), and a² + b² = c², so that side is c.
- The two triangles have the same three sides: a, b and c. So they are the same triangle (congruent), and the angle opposite c in the first one is the right angle of the second.
- Keep a and b fixed and open the corner between them. The third side gets longer as the angle grows, and it is exactly c when the angle is 90°. So a longer third side (c² more than a² + b²) means an obtuse angle, and a shorter one means an acute angle.
Two slips to watch for. Always put the longest side on the c² side: for 5, 12, 13 the test is 5² + 12² against 13², not 5² + 13² against 12². And “nearly equal” is not equal: 9, 12, 16 gives 81 + 144 = 225 against 256, and that corner is not 90 degrees.
Worked examples
The rope: sides of 3, 4 and 5 gaps. Is the corner a right angle?
- The longest side is 5, so c = 5.
- 3² + 4² = 9 + 16 = 25, and 5² = 25.
- They are equal, so by the converse the corner is a right angle. The carpenter was right.
A triangle has sides 7, 8 and 11. Is it right-angled? If not, what type of angle is the largest?
- The longest side is 11, so c = 11.
- 7² + 8² = 49 + 64 = 113, and 11² = 121.
- 113 is less than 121, so it is not right-angled: the angle opposite 11 is obtuse.
Your go. The first step is given. Is a triangle with sides 20, 21 and 29 right-angled?
- The longest side is 29, so c = 29 and the two shorter sides are 20 and 21.
- Work out 20² + 21² and 29², compare them, then check.
Check the answer
400 + 441 = 841 and 29² = 841. They are equal, so it is right-angled.
With a calculator
You do not need a square root key to test a triangle: compare a² + b² with c². Work out each side's square with the square key, add the two short ones, and compare. Two numbers that are very close, such as 101 and 100, are still not equal, so a triangle whose squares gave those two numbers is not right-angled.
Check before you go on
1. A triangle has sides 6, 8 and 11. Is it right-angled?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. A triangle has sides 12, 35 and 37. Work out 12² + 35².
3. Which set of three lengths could be the sides of a right-angled triangle?
Level 2 · Building
4. The sides 9, 40 and x make a right-angled triangle, with x the longest side. What is x?
5. How many of these sets of lengths are right-angled triangles? (9, 12, 15) (11, 60, 61) (8, 14, 17) (7, 9, 12) (16, 63, 65)
6. A carpenter checks a door frame that is 120 cm wide and 90 cm high. How long should the diagonal be for the corners to be right angles, in cm?
Level 3 · Stretch
7. 5, 12, 13 is a right-angled triangle. A smaller triangle has sides 2.5, 6 and x, with the same shape. What is x?
8. A right-angled triangle has sides x, x + 7 and 13, with 13 the longest side. What is x?
Something to think about
3, 4, 5 is a right-angled triangle. A student says that 3.1, 4.1, 5.1 is one too, because each side is a bit bigger. Is she right? Use the figure or the theorem.
One way to think about it
No. 3.1² + 4.1² = 9.61 + 16.81 = 26.42, but 5.1² = 26.01. The two are not equal, so the corner is not exactly 90 degrees (it is acute). Adding the same amount to every side changes the shape. Multiplying every side by the same number does not.
Exit ticket
9. A triangle has sides 11, 13 and 17. Which statement is correct?