Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- Work out 8² + 15².
Answer
64 + 225 = 289. - What is √144?
Answer
12, because 12 × 12 = 144. - Which side is the hypotenuse of a right-angled triangle?
Answer
The side opposite the right angle. It is the longest side.
Words used in this lesson
- hypotenuse
- the longest side of a right-angled triangle, opposite the right angle. For example: 13 m in the ladder triangle.
- leg
- one of the two shorter sides that meet at the right angle. For example: 5 m along the ground.
- square (a number)
- a number multiplied by itself. For example: the square of 5 is 25.
- square root
- the number that squares to give a number. For example: the square root of 144 is 12.
- diagonal
- a straight line joining opposite corners. For example: the diagonal of a screen.
Why a ladder is not 13 − 5
A 13 m ladder leans against a wall with its foot 5 m from the wall. Does the top reach higher or lower than 8 m, which is 13 − 5? Make a guess before you read on.
In a right-angled triangle, a² + b² = c², where c is the hypotenuse (the side opposite the right angle, and always the longest side) and a and b are the other two sides.
Why it works: the square on the hypotenuse has exactly the area of the other two squares put together; the proof is below the first figure. To find a shorter side, subtract: a² = c² − b².
Set the two legs and read the three areas. Before each move, predict c.
a² + b² = 9 + 16 = 25, so c = √25 = 5.
Show me the steps
- a = 3, b = 4: 9 + 16 = 25, so c = 5.
- a = 6, b = 8: 36 + 64 = 100, so c = 10. Doubling both legs doubles c.
- a = 2, b = 3: 4 + 9 = 13, so c = √13 ≈ 3.61. The hypotenuse need not be a whole number.
- a = 5, b = 5: 25 + 25 = 50, so c = √50 ≈ 7.07. It is longer than each leg, but shorter than 5 + 5 = 10.
Why it is true
Trying 3, 4, 5 does not prove the theorem (Lesson 1). A proof has to work for any right-angled triangle. Press the two buttons and watch what changes, and what does not.
Arrangement 1: the long sides of the four triangles make a tilted square. The empty space is c², and it is 49 − 24 = 25.
Show me the steps
- Arrangement 1: the four triangles sit in the corners of a square of side 3 + 4 = 7. Their long sides make a tilted square of side c, and the empty space is c².
- Arrangement 2: the same four triangles sit in the other way. The empty space is now a 3 by 3 square and a 4 by 4 square, so it is a² + b² = 9 + 16.
- The big square is 49 and the four triangles are 4 × 6 = 24 in both, so the empty space is 49 − 24 = 25 in both. That is c² = a² + b².
Proof that a² + b² = c² in every right-angled triangle.
- Take four copies of the triangle, with legs a and b and hypotenuse c. Fit them inside a square of side a + b, whose area is (a + b)². The four triangles have area 4 × ½ab = 2ab together.
- Arrangement 1: the hypotenuses form a tilted square of side c. (The two acute angles of a right-angled triangle add to 90°, so each corner of the tilted shape is 180° − 90° = 90°.) The empty space is c².
- Arrangement 2: the same four triangles leave a square of side a and a square of side b. The empty space is a² + b².
- Both arrangements are the same big square minus the same four triangles, so the empty spaces are equal: c² = a² + b². Nothing in this argument depends on which a and b we chose, so it is a proof, not a test.
Two slips to watch for. c is not a + b: for legs 3 and 4 the hypotenuse is 5, not 7. And a² + b² gives c², not c, so finish with the square root. The theorem is only true when one angle is a right angle.
Worked examples
The ladder: 13 m long, foot 5 m from the wall. How high does it reach?
- The wall is vertical and the ground horizontal, so the ladder is the hypotenuse: c = 13, and one leg is 5.
- The unknown is a shorter side, so subtract: a² = 13² − 5² = 169 − 25 = 144.
- a = √144 = 12 m. That is higher than 8 m, because the ladder is the long side, not the difference of the other two.
A support wire 17 m long is fixed to a pole and pegged 8 m from the pole's foot. How tall is the pole?
- The wire is the hypotenuse (17), and the distance on the ground is a leg (8).
- Pole² = 17² − 8² = 289 − 64 = 225.
- Pole = √225 = 15 m.
Your go. The first step is given. A rectangular tile is 12 cm by 7 cm. How long is its diagonal? Give the answer to 1 decimal place.
- The diagonal splits the rectangle into two right-angled triangles with legs 12 and 7. The diagonal is the hypotenuse.
- Square, add, then take the root. Then check.
Check the answer
144 + 49 = 193, and √193 = 13.9 cm to 1 decimal place.
With a calculator
Do the squares and the addition first, then take the square root of the whole total. For legs 8 and 15, press 8 and the square key, add 15 and the square key, then the square root key on the result: 17. Pressing the square root key too early gives √64 + 225 = 8 + 225 = 233.
Check that the answer is sensible: the hypotenuse must be longer than each leg and shorter than the legs added.
Check before you go on
1. A right-angled triangle has legs 7 and 10 and hypotenuse c. Which equation is correct?
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 right-angled triangle has legs 9 cm and 12 cm. How long is the hypotenuse, in cm?
3. A right-angled triangle has legs 4 and 9. What is the area of the square on the hypotenuse?
Level 2 · Building
4. The hypotenuse of a right-angled triangle is 25 cm and one leg is 15 cm. How long is the other leg, in cm?
5. A tablet screen is 24 cm wide and 18 cm high. How long is its diagonal, in cm?
6. A sheet of card is 21 cm by 12 cm. How long is its diagonal, to 1 decimal place?
Level 3 · Stretch
7. A drone flies 36 m due east, then 15 m due north. How far is it from where it started, in metres?
8. An isosceles triangle has two sides of 25 cm and a base of 14 cm. How tall is it, measured from the base to the top, in cm?
Something to think about
The two legs are 5 and 12, and the hypotenuse is 13. If you add 1 to each side, giving 6, 13 and 14, is it still a right-angled triangle? Check with the theorem.
One way to think about it
No. 6² + 13² = 36 + 169 = 205, but 14² = 196. The theorem only holds for sides that fit exactly, so adding the same amount to every side breaks it. Multiplying every side by the same number keeps it (5, 12, 13 becomes 10, 24, 26).
Exit ticket
9. The hypotenuse is 20 and one leg is 12. Which calculation finds the other leg?