Chapter 2 of 8

All 8 chapters
  1. 1Pythagoras' theorem, readunderstood
  2. 2Using Pythagoras, readunderstood
  3. 3Opposite, adjacent and hypotenuse, readunderstood
  4. 4Sine, cosine and tangent, readunderstood
  5. 5Finding angles, readunderstood
  6. 6Exact values, readunderstood
  7. 7The sine rule and the cosine rule, readunderstood
  8. 8Area of any triangle, and bearings, readunderstood

M4.3.2 Chapter 2

Using Pythagoras

A box is 2 cm by 3 cm by 6 cm. How long is the longest stick that fits inside it?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Find how high a ladder reaches up a wall
  • Find the distance between two points on a grid
  • Find the longest stick that fits inside a box

A wall stands straight up from flat ground. So the wall and the ground meet at a right angle. A ladder leaning on the wall makes a right angled triangle with them. The ladder is the hypotenuse.

Draw it first

Find the triangle in the ladder

  1. Step 1: a ladder leaning on a wall, with the height labelled h, the foot distance 0.9 m and the ladder 4.1 m. Step 2: the same measurements as a right angled triangle ABC, with the sides labelled h, 0.9 m and 4.1 m.0.9 mh4.1 m

    1 A ladder leans on a wall. Its foot is 0.9 m from the wall. Its top reaches a height h.

  2. Step 1: a ladder leaning on a wall, with the height labelled h, the foot distance 0.9 m and the ladder 4.1 m. Step 2: the same measurements as a right angled triangle ABC, with the sides labelled h, 0.9 m and 4.1 m.0.9 mh4.1 mABCh0.9 m4.1 m

    2 Draw the wall, the ground and the ladder as a triangle. The ladder, 4.1 m, is the hypotenuse.

You know the hypotenuse and one leg, so you subtract the squares. The height is h² = 4.1² − 0.9².

Worked example

A ladder 4.1 m long stands with its foot 0.9 m from a wall. How high up the wall does it reach?

  1. The ladder is the hypotenuse: c = 4.1. The foot is b = 0.9 from the wall. We want the height h.
  2. Use h² = c² − b² = 4.1² − 0.9².
  3. Square each number: 4.1² = 16.81 and 0.9² = 0.81.
  4. Subtract: h² = 16.81 − 0.81 = 16.
  5. Take the square root: h = √16 = 4.

Answer: The ladder reaches 4 m up the wall.

A grid hides a right angled triangle too. To find how far apart two points are, go across, then go up. The two moves are the legs. The straight line between the points is the hypotenuse.

A map grid with the axis east (km) along the bottom and the axis north (km) up the side. Point A is at (1, 2) and point B is at (7, 10). A heavy line d joins them. A dashed line labelled 6 km runs east from A, and a dashed line labelled 8 km runs north to B.ABd6 km8 km1234567892468100east (km)north (km)
A is at (1, 2) and B is at (7, 10). One square is 1 km.

Worked example

A walker goes from A (1, 2) to B (7, 10) on a map. One square is 1 km. How far apart are A and B in a straight line?

  1. Across from A to B: 7 − 1 = 6 km east.
  2. Up from A to B: 10 − 2 = 8 km north.
  3. The two moves are the legs of a right angled triangle. The distance d is its hypotenuse.
  4. d² = 6² + 8² = 36 + 64 = 100.
  5. d = √100 = 10 km.

Answer: The distance is 10 km.

Take care when a coordinate is negative. To go from −2 to 3 you move 3 − (−2) = 5 places. You always take the smaller number away from the larger one.

A map grid with the axes crossing in the middle. The axis along the bottom is labelled x and the axis up the side is labelled y. Point P is at (−2, −3) and point Q is at (3, 2). A heavy line labelled d joins them. A dashed line labelled 5 km runs across from P to the corner (3, −3), and a dashed line labelled 5 km runs up from that corner to Q.PQd5 km5 km−224−4−220xy
P is at (−2, −3) and Q is at (3, 2). One square is 1 km, and x and y are the kilometres east and north.

Worked example

Two huts are at P (−2, −3) and Q (3, 2) on a map. One square is 1 km. How far apart are they, to 1 decimal place?

  1. Across from P to Q: 3 − (−2) = 5 km.
  2. Up from P to Q: 2 − (−3) = 5 km.
  3. The two moves are the legs of a right angled triangle. The distance d is its hypotenuse.
  4. d² = 5² + 5² = 25 + 25 = 50.
  5. d = √50 = 7.071 on the calculator. To 1 decimal place that is 7.1.

Answer: The huts are 7.1 km apart, to 1 decimal place.

A box has a longest stick too. It runs from a bottom corner to the opposite top corner. That line is the space diagonal. We find it with two right angled triangles, one flat on the floor and one standing up.

A box 4 cm long, 3 cm wide and 12 cm high. A dashed line x runs across the floor from one corner to the opposite corner. A heavy line d runs from the same bottom corner to the opposite top corner. The edges are labelled 12 cm, 3 cm and 4 cm.4 cm12 cm3 cmxd
The box is drawn from the side, so the depth looks shorter than it is. The labels give the true lengths.
Two right angled triangles side by side, drawn to the same scale. The left one lies on the floor, with legs 4 cm and 3 cm and the hypotenuse x. The right one stands up, with legs x and 12 cm and the hypotenuse d. The corners are named A, B and C in each.ABC3 cm4 cmxABC12 cmxd
Triangle one finds x. Triangle two, built on x, finds d.

Worked example

A box is 4 cm long, 3 cm wide and 12 cm high. How long is the longest stick that fits inside?

  1. The floor is a rectangle 4 cm by 3 cm. Its diagonal x is the hypotenuse of a right angled triangle.
  2. x² = 4² + 3² = 16 + 9 = 25.
  3. x = √25 = 5 cm.
  4. Now stand a triangle on that diagonal. Its legs are x and the height, and the stick d is its hypotenuse.
  5. d² = 5² + 12² = 25 + 144 = 169.
  6. d = √169 = 13 cm.

Answer: The longest stick is 13 cm.

Look at the two sums together: d² = x² + 12², and x² = 4² + 3². So d² = 4² + 3² + 12². For any box, d² = l² + w² + h². You can go straight to it, but you must understand where it comes from.

See it three ways

With things

Lean a pencil in a box so it touches a bottom corner and the opposite top corner. It is the longest pencil the box holds.

As a picture

A box 8 cm long, 4 cm wide and 1 cm high with a heavy line d from one bottom corner to the opposite top corner. The edges are labelled 8 cm, 4 cm and 1 cm.8 cm1 cm4 cmd

In symbols

d² = l² + w² + h², so d = √(l² + w² + h²).

Try it

Measure a diagonal, then check it

You need: a sheet of paper, a ruler.

  1. Measure the long side and the short side of the sheet. Say they are 30 cm and 20 cm.
  2. Work out 30² + 20², then take the square root.
  3. Measure the diagonal, corner to corner. Compare.

What you will see: For 30 cm and 20 cm, the diagonal is √1 300, which is 36.06 cm. To the nearest centimetre it is 36 cm. Your measure should be close to the sum.

If you cannot try it, this is what happens.
A right angled triangle made by a sheet of paper cut along its diagonal. The legs are labelled 30 cm and 20 cm. The hypotenuse is labelled 36 cm. The corners are named A, B and C.ABC20 cm30 cm36 cm

A wrong idea many people have

The distance between two points is the across move plus the up move.

Why it feels right: On a grid you count squares along the lines, and counting is adding.

The test that breaks it: For A (1, 2) and B (7, 10), across plus up is 6 + 8 = 14. Measure A to B with the squares of the grid: it is 10 squares, so 10 km.

What is true: Across plus up is the way along the two legs. The straight line is the hypotenuse, which is shorter: d = √(6² + 8²) = 10.

A wrong idea many people have

The longest stick in a box is as long as its longest edge.

Why it feels right: A stick lying along the longest edge looks as long as the box allows.

The test that breaks it: In the 4 cm by 3 cm by 12 cm box, the longest edge is 12 cm. But a 13 cm stick fits when you slant it corner to corner, because √169 = 13.

What is true: The corner to corner line through the middle is longer than any edge. d = √(l² + w² + h²).

Key words

space diagonal
the line from one corner of a box to the opposite corner, through the inside.

Back to your guess

The answer is C: 7 cm, from one bottom corner to the opposite top corner.

The longest stick runs corner to corner through the middle, and it is 7 cm. The sum is 2² + 3² + 6² = 4 + 9 + 36 = 49, and √49 = 7. A stick cannot bend, so 11 cm is too long. This chapter uses right angled triangles to find such lengths.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. A ladder 7.5 m long stands with its foot 2.1 m from a wall. How high up the wall does it reach?

    A ladder leaning on a wall. The ladder is labelled 7.5 m, the foot distance is labelled 2.1 m and the height is labelled h.2.1 mh7.5 m
    Tap your answer.

    1. Not this one. You added the squares, 7.5² + 2.1². The ladder is the hypotenuse, so it is the longest side, and 7.8 m is more than the ladder. Subtract to find the height. Go to M4.3.1. Try another.

    2. Not this one. You took 2.1 from 7.5. Lengths do not subtract like that in a right angled triangle. Subtract the squares, then take the root. Go to M4.3.1. Try another.

    3. Yes, that is right. h² = 7.5² − 2.1² = 56.25 − 4.41 = 51.84, and √51.84 = 7.2.

    4. Not this one. That is h², the height squared: 56.25 − 4.41. Take the square root to get h. Go to M4.3.1. Try another.

    See the answer

    7.2 m

    h² = 7.5² − 2.1² = 56.25 − 4.41 = 51.84, and √51.84 = 7.2.

    • If you chose A: You added the squares, 7.5² + 2.1². The ladder is the hypotenuse, so it is the longest side, and 7.8 m is more than the ladder. Subtract to find the height. Go to M4.3.1.
    • If you chose B: You took 2.1 from 7.5. Lengths do not subtract like that in a right angled triangle. Subtract the squares, then take the root. Go to M4.3.1.
    • If you chose D: That is h², the height squared: 56.25 − 4.41. Take the square root to get h. Go to M4.3.1.

  2. On a map, hut P is at (1, 1) and hut Q is at (13, 6). One square is 1 km. How far apart are the huts?

    A map grid with the axis east (km) along the bottom and the axis north (km) up the side. Point P is at (1, 1) and point Q is at (13, 6). A heavy line joins them.PQ24681012142460east (km)north (km)
    Tap your answer.

    1. Yes, that is right. Across: 13 − 1 = 12. Up: 6 − 1 = 5. d² = 12² + 5² = 144 + 25 = 169, and √169 = 13.

    2. Not this one. You added the across move and the up move, 12 + 5. That is the way along the grid lines. The straight line is shorter. Go to M4.3.2. Try another.

    3. Not this one. You took 5 from 12. Both moves are legs of the triangle, so you square both and add. Go to M4.3.1. Try another.

    4. Not this one. 169 is d², the distance squared: 144 + 25. Take the square root to get d. Go to M4.3.1. Try another.

    See the answer

    13 km

    Across: 13 − 1 = 12. Up: 6 − 1 = 5. d² = 12² + 5² = 144 + 25 = 169, and √169 = 13.

    • If you chose B: You added the across move and the up move, 12 + 5. That is the way along the grid lines. The straight line is shorter. Go to M4.3.2.
    • If you chose C: You took 5 from 12. Both moves are legs of the triangle, so you square both and add. Go to M4.3.1.
    • If you chose D: 169 is d², the distance squared: 144 + 25. Take the square root to get d. Go to M4.3.1.

  3. On a map, A is at (−5, −2) and B is at (3, 4). One square is 1 km. How far apart are A and B?

    A map grid with the axes crossing in the middle. The axis along the bottom is labelled x and the axis up the side is labelled y. Point A is at (−5, −2) and point B is at (3, 4). A heavy line joins them.AB−6−4−224−4−2240xy
    Tap your answer.

    1. Not this one. You worked out 3 − 5 and 4 − 2, which drops the minus signs. Count the places instead: from −5 to 3 is 8, and from −2 to 4 is 6. Go to M4.3.2. Try another.

    2. Not this one. You added the two moves, 8 + 6. That is the way along the grid lines. The straight line is the hypotenuse, which is shorter. Go to M4.3.2. Try another.

    3. Not this one. 100 is d², the distance squared: 64 + 36. Take the square root to get d. Go to M4.3.1. Try another.

    4. Yes, that is right. Across: 3 − (−5) = 8. Up: 4 − (−2) = 6. d² = 8² + 6² = 64 + 36 = 100, and √100 = 10.

    See the answer

    10 km

    Across: 3 − (−5) = 8. Up: 4 − (−2) = 6. d² = 8² + 6² = 64 + 36 = 100, and √100 = 10.

    • If you chose A: You worked out 3 − 5 and 4 − 2, which drops the minus signs. Count the places instead: from −5 to 3 is 8, and from −2 to 4 is 6. Go to M4.3.2.
    • If you chose B: You added the two moves, 8 + 6. That is the way along the grid lines. The straight line is the hypotenuse, which is shorter. Go to M4.3.2.
    • If you chose C: 100 is d², the distance squared: 64 + 36. Take the square root to get d. Go to M4.3.1.

  4. A flat box is 8 cm long, 4 cm wide and 1 cm high. How long is the longest stick that fits inside?

    A flat box 8 cm long, 4 cm wide and 1 cm high with a heavy line d from one bottom corner to the opposite top corner. The edges are labelled 8 cm, 4 cm and 1 cm.8 cm1 cm4 cmd
    Tap your answer.

    1. Not this one. That is the longest edge. A stick slanted corner to corner is longer than any edge. Go to M4.3.2. Try another.

    2. Yes, that is right. d² = 8² + 4² + 1² = 64 + 16 + 1 = 81, and √81 = 9.

    3. Not this one. You added the three edges, 8 + 4 + 1. A stick cannot bend round, so it must be shorter than that. Go to M4.3.2. Try another.

    4. Not this one. That is the diagonal of the floor, √(8² + 4²). The stick also rises 1 cm, so it is a little longer. Use all three edges. Go to M4.3.2. Try another.

    See the answer

    9 cm

    d² = 8² + 4² + 1² = 64 + 16 + 1 = 81, and √81 = 9.

    • If you chose A: That is the longest edge. A stick slanted corner to corner is longer than any edge. Go to M4.3.2.
    • If you chose C: You added the three edges, 8 + 4 + 1. A stick cannot bend round, so it must be shorter than that. Go to M4.3.2.
    • If you chose D: That is the diagonal of the floor, √(8² + 4²). The stick also rises 1 cm, so it is a little longer. Use all three edges. Go to M4.3.2.

  5. A ladder stands with its foot some distance from a wall. You know the length of the ladder and the foot distance. How do you find the height it reaches?

    Tap your answer.

    1. Yes, that is right. The ladder is the hypotenuse, so ladder² = height² + foot². Take foot² from both sides to get height² = ladder² − foot².

    2. Not this one. The ladder is the hypotenuse, the longest side. The squares of the legs add up to its square. To find a leg you subtract. Go to M4.3.1. Try another.

    3. Not this one. Lengths in a right angled triangle do not subtract like that. A ladder of 5 m with its foot 3 m out reaches 4 m, not 2 m. Subtract the squares. Go to M4.3.1. Try another.

    See the answer

    Square the ladder length, take away the square of the foot distance, then take the square root.

    The ladder is the hypotenuse, so ladder² = height² + foot². Take foot² from both sides to get height² = ladder² − foot².

    • If you chose B: The ladder is the hypotenuse, the longest side. The squares of the legs add up to its square. To find a leg you subtract. Go to M4.3.1.
    • If you chose C: Lengths in a right angled triangle do not subtract like that. A ladder of 5 m with its foot 3 m out reaches 4 m, not 2 m. Subtract the squares. Go to M4.3.1.

  6. From an earlier chapter

    Which set of three sides, in cm, makes a right angled triangle?

    Tap your answer.

    1. Not this one. Test it: 6² + 8² = 36 + 64 = 100, but 9² = 81. They are not equal. Go to M4.3.1. Try another.

    2. Not this one. Test it: 9² + 12² = 81 + 144 = 225, but 16² = 256. They are not equal. Go to M4.3.1. Try another.

    3. Yes, that is right. 7² + 24² = 49 + 576 = 625 and 25² = 625. The two are equal, so the triangle has a right angle.

    4. Not this one. Test it: 5² + 7² = 25 + 49 = 74, but 10² = 100. They are not equal. Go to M4.3.1. Try another.

    See the answer

    7, 24, 25

    7² + 24² = 49 + 576 = 625 and 25² = 625. The two are equal, so the triangle has a right angle.

    • If you chose A: Test it: 6² + 8² = 36 + 64 = 100, but 9² = 81. They are not equal. Go to M4.3.1.
    • If you chose B: Test it: 9² + 12² = 81 + 144 = 225, but 16² = 256. They are not equal. Go to M4.3.1.
    • If you chose D: Test it: 5² + 7² = 25 + 49 = 74, but 10² = 100. They are not equal. Go to M4.3.1.

  7. Two huts are at (2, 3) and (9, 27) on a map. One square is 1 km. How many kilometres apart are they?

    See the answer

    25

    Across: 7. Up: 24. d² = 7² + 24² = 49 + 576 = 625, and √625 = 25.

  8. A box is 2 cm long, 4 cm wide and 4 cm high. How many centimetres long is the longest stick that fits inside?

    See the answer

    6

    d² = 2² + 4² + 4² = 4 + 16 + 16 = 36, and √36 = 6.

  9. A ladder 5 m long stands with its foot 1.4 m from a wall. How many metres up the wall does it reach?

    See the answer

    4.8

    h² = 5² − 1.4² = 25 − 1.96 = 23.04, and √23.04 = 4.8.

Keep it

Copy this into your exercise book.

Spot the right angled triangle. A ladder is the hypotenuse. Across and up on a grid are the legs. In a box, d² = l² + w² + h².

Draw: A box with its floor diagonal x dashed and its space diagonal d heavy, and d² = l² + w² + h² under it.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

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