Chapter 3 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

M4.3.3 Chapter 3

Opposite, adjacent and hypotenuse

Two right angled triangles have the same angles, but one is twice as big. Is opposite ÷ adjacent bigger in the big one?

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

  • Name the opposite, adjacent and hypotenuse for a given angle
  • Show by drawing that opposite ÷ adjacent is fixed for each angle

In Chapter 1 the sides had names that did not depend on an angle. Now we pick one of the two acute angles and call it θ, the Greek letter theta. The names of two sides then depend on where θ is.

The hypotenuse is still the side opposite the right angle. The opposite side faces θ: it is across from θ and does not touch it. The adjacent side is next to θ: it touches θ and is not the hypotenuse.

Draw it first

Name the sides for the angle θ

  1. Step 1: a right angled triangle ABC with a small square at C and an angle θ marked at A. The corners are named A, B and C. Step 2: the sides are named. The side across from θ is labelled opposite, the side next to θ is labelled adjacent, and the side facing the right angle is labelled hypotenuse.ABCθ

    1 Pick an acute angle and mark it θ. The right angle has its small square.

  2. Step 1: a right angled triangle ABC with a small square at C and an angle θ marked at A. The corners are named A, B and C. Step 2: the sides are named. The side across from θ is labelled opposite, the side next to θ is labelled adjacent, and the side facing the right angle is labelled hypotenuse.ABCθoppositeadjacenthypotenuse

    2 Name the sides: hypotenuse faces the right angle, opposite faces θ, adjacent touches θ.

Choose the other angle and the names swap. The side that was opposite becomes adjacent, and the other way round. The hypotenuse never changes.

The same right angled triangle ABC, now with θ marked at B. The side AC, across from θ, is labelled opposite. The side BC, next to θ, is labelled adjacent. The side AB is labelled hypotenuse.ABCadjacentoppositehypotenuseθ
The same triangle. θ is at B now, so opposite and adjacent swap.

Worked example

In triangle PQR the right angle is at Q. PQ is 5 cm, QR is 12 cm and PR is 13 cm. The angle θ is at P. Name the sides and find opposite ÷ adjacent.

  1. The right angle is at Q. The hypotenuse faces it: PR = 13 cm.
  2. θ is at P. The opposite side is across from P: QR = 12 cm.
  3. The adjacent side touches P and is not the hypotenuse: PQ = 5 cm.
  4. Opposite ÷ adjacent = 12 ÷ 5 = 2.4.

Answer: Opposite ÷ adjacent = 12 ÷ 5 = 2.4.

A right angled triangle PQR with a small square at Q. The angle θ is marked at P. The side PQ is labelled 5 cm, the side QR is labelled 12 cm and the side PR is labelled 13 cm.PQR12 cm13 cm5 cmθ

Now the big idea. Draw an angle θ. Stand a vertical line at 4 cm, at 8 cm and at 12 cm along the flat arm. You get three right angled triangles. They all have the same θ.

Three right angled triangles that share the angle θ and one slanted line. The flat sides are labelled 4 cm, 8 cm and 12 cm. The upright sides are labelled 3 cm, 6 cm and 9 cm.4 cm3 cm8 cm6 cm12 cm9 cmθ
One angle θ, three triangles. Each is the one before, made bigger.
A table with the head Opp cm, Adj cm and Opp ÷ adj. Row 1: 3, 4, 0.75. Row 2: 6, 8, 0.75. Row 3: 9, 12, 0.75. Opp is the opposite side and Adj is the adjacent side.Opp cmAdj cmOpp ÷ adj340.75680.759120.75
The last column never changes.

Why? Triangle 2 is triangle 1 enlarged by a factor of 2: every side is doubled. Triangle 3 is an enlargement by a factor of 3. When all sides are multiplied by the same number, a division of two sides stays the same. So for one angle, opposite ÷ adjacent is a fixed number. Change the angle and the number changes.

We can use the fixed number. If we know θ gives 0.75, then in any right angled triangle with that θ, the opposite side is 0.75 times the adjacent side.

Two right angled triangles that share the angle θ. The small one has a flat side of 4 cm and an upright side of 3 cm. The big one has a flat side of 12 cm and an upright side labelled x.4 cm3 cm12 cmxθ

Worked example

A small triangle has opposite 3 cm and adjacent 4 cm for θ. A bigger triangle has the same θ and adjacent 12 cm. Find its opposite side x.

  1. In the small triangle, opposite ÷ adjacent = 3 ÷ 4 = 0.75.
  2. The big triangle has the same angle θ, so its opposite ÷ adjacent is also 0.75.
  3. So x ÷ 12 = 0.75.
  4. Multiply both sides by 12: x = 0.75 × 12 = 9.

Answer: The opposite side is 9 cm.

See it three ways

With things

Two ramps with the same steepness. The short one goes 4 m along and rises 3 m. The long one goes 8 m along and rises 6 m.

As a picture

Two right angled triangles that share the angle θ. The flat sides are labelled 4 m and 8 m. The upright sides are labelled 3 m and 6 m.4 m3 m8 m6 mθ

In symbols

3 ÷ 4 = 6 ÷ 8 = 0.75.

Try it

Draw two triangles with one angle

You need: paper, a ruler, a protractor, a pencil.

  1. Draw a flat line 10 cm long. At its left end, use the protractor to draw a line at 40° above it.
  2. Stand a vertical line at 5 cm and another at 10 cm along the flat line, up to the slanted line.
  3. Measure the two vertical lines. Divide each by its flat length.

What you will see: The vertical lines measure about 4.2 cm and 8.4 cm. Both divisions give about 0.84. A bigger triangle, the same answer.

If you cannot try it, this is what happens.
Two right angled triangles that share an angle of 40°. The flat sides are labelled 5 cm and 10 cm. The upright sides are labelled 4.2 cm and 8.4 cm.5 cm4.2 cm10 cm8.4 cm40°

A wrong idea many people have

A bigger triangle with the same angle has a bigger opposite ÷ adjacent.

Why it feels right: Both the opposite side and the adjacent side get longer, and we think a longer side means a bigger answer.

The test that breaks it: Take 3 ÷ 4 = 0.75 and 6 ÷ 8 = 0.75. The two sides doubled, and the answer stayed at 0.75.

What is true: Doubling both sides leaves the division the same. Opposite ÷ adjacent depends only on the angle.

A wrong idea many people have

The adjacent side is always the one along the bottom.

Why it feels right: In most pictures the flat side is called adjacent, and a position is quicker to remember than a rule.

The test that breaks it: Turn the triangle so another side is on the bottom. Opposite and adjacent have not moved: they still face and touch θ.

What is true: Opposite and adjacent depend on where θ is, not on how the page is turned. Look for the side that touches θ and is not the hypotenuse.

Key words

opposite
the side across from the angle θ, not touching it.
adjacent
the side next to the angle θ that is not the hypotenuse.

Back to your guess

The answer is A: No, it is the same, because both sides are multiplied by the same number.

The answer is the same. In the big triangle both sides are twice as long, and 6 ÷ 8 = 3 ÷ 4. For a given angle, opposite ÷ adjacent never changes. That fixed number is the start of trigonometry.

Check yourself

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

  1. The angle θ is at R. Which side is opposite θ?

    A right angled triangle PQR with a small square at Q and the angle θ marked at R. The corners are named P, Q and R.PQRθ
    Tap your answer.

    1. Yes, that is right. θ is at R. The side across from R is PQ, and it does not touch R. So PQ is opposite θ.

    2. Not this one. QR touches R, where θ is, and it is not the longest side. So QR is the adjacent side. Go to M4.3.3. Try another.

    3. Not this one. PR faces the right angle at Q. That makes it the hypotenuse, not the opposite side. Go to M4.3.3. Try another.

    4. Not this one. Turning the page changes nothing. The opposite side is the one across from θ, whichever way the triangle sits. Go to M4.3.3. Try another.

    See the answer

    PQ

    θ is at R. The side across from R is PQ, and it does not touch R. So PQ is opposite θ.

    • If you chose B: QR touches R, where θ is, and it is not the longest side. So QR is the adjacent side. Go to M4.3.3.
    • If you chose C: PR faces the right angle at Q. That makes it the hypotenuse, not the opposite side. Go to M4.3.3.
    • If you chose D: Turning the page changes nothing. The opposite side is the one across from θ, whichever way the triangle sits. Go to M4.3.3.

  2. Now θ is at P instead. Which side is adjacent to θ?

    The same right angled triangle PQR with a small square at Q, now with the angle θ marked at P. The corners are named P, Q and R.PQRθ
    Tap your answer.

    1. Not this one. QR is across from P, so it is the opposite side now. The sides swap names when θ moves. Go to M4.3.3. Try another.

    2. Not this one. PR touches P, but it is the hypotenuse. The adjacent side touches θ and is not the hypotenuse. Go to M4.3.3. Try another.

    3. Yes, that is right. θ is at P. Two sides touch P: PQ and PR. PR faces the right angle, so it is the hypotenuse. That leaves PQ as the adjacent side.

    See the answer

    PQ

    θ is at P. Two sides touch P: PQ and PR. PR faces the right angle, so it is the hypotenuse. That leaves PQ as the adjacent side.

    • If you chose A: QR is across from P, so it is the opposite side now. The sides swap names when θ moves. Go to M4.3.3.
    • If you chose B: PR touches P, but it is the hypotenuse. The adjacent side touches θ and is not the hypotenuse. Go to M4.3.3.

  3. A right angled triangle has opposite 6 cm, adjacent 8 cm and hypotenuse 10 cm for the angle θ. What is opposite ÷ adjacent?

    A right angled triangle ABC with a small square at C and the angle θ marked at A. The side BC is labelled opposite 6 cm, the side AC is labelled adjacent 8 cm and the side AB is labelled 10 cm.ABCopposite 6 cmadjacent 8 cm10 cmθ
    Tap your answer.

    1. Not this one. You divided adjacent by opposite, 8 ÷ 6. The question asks for opposite ÷ adjacent, in that order. Go to M4.3.3. Try another.

    2. Yes, that is right. Opposite ÷ adjacent = 6 ÷ 8 = 0.75.

    3. Not this one. You divided the opposite by the hypotenuse, 6 ÷ 10. That is a different ratio. Here the bottom number is the adjacent side. Go to M4.3.3. Try another.

    4. Not this one. You divided the adjacent by the hypotenuse, 8 ÷ 10. The question uses opposite on top and adjacent underneath. Go to M4.3.3. Try another.

    See the answer

    0.75

    Opposite ÷ adjacent = 6 ÷ 8 = 0.75.

    • If you chose A: You divided adjacent by opposite, 8 ÷ 6. The question asks for opposite ÷ adjacent, in that order. Go to M4.3.3.
    • If you chose C: You divided the opposite by the hypotenuse, 6 ÷ 10. That is a different ratio. Here the bottom number is the adjacent side. Go to M4.3.3.
    • If you chose D: You divided the adjacent by the hypotenuse, 8 ÷ 10. The question uses opposite on top and adjacent underneath. Go to M4.3.3.

  4. Triangle A has opposite 3 cm and adjacent 5 cm for θ. Triangle B has the same θ and adjacent 15 cm. How long is the opposite side of B?

    Tap your answer.

    1. Not this one. You added 10, the growth of the adjacent side, to the opposite. The sides grow by multiplying, not adding: 5 became 15, which is 3 times bigger. Go to M4.3.3. Try another.

    2. Not this one. That is the adjacent side of A. In B the opposite side grows with the rest of the triangle. Go to M4.3.3. Try another.

    3. Not this one. You multiplied 3 by 15. The ratio 3 ÷ 5 is fixed, so the opposite is 3 ÷ 5 of 15, which is smaller than 15. Go to M4.3.3. Try another.

    4. Yes, that is right. B is A made 3 times bigger. So the opposite is 3 × 3 = 9. Check: 9 ÷ 15 = 3 ÷ 5.

    See the answer

    9 cm

    B is A made 3 times bigger. So the opposite is 3 × 3 = 9. Check: 9 ÷ 15 = 3 ÷ 5.

    • If you chose A: You added 10, the growth of the adjacent side, to the opposite. The sides grow by multiplying, not adding: 5 became 15, which is 3 times bigger. Go to M4.3.3.
    • If you chose B: That is the adjacent side of A. In B the opposite side grows with the rest of the triangle. Go to M4.3.3.
    • If you chose C: You multiplied 3 by 15. The ratio 3 ÷ 5 is fixed, so the opposite is 3 ÷ 5 of 15, which is smaller than 15. Go to M4.3.3.

  5. Two right angled triangles have the same angle θ. One is 3 times as big as the other. What is the same in both?

    Tap your answer.

    1. Yes, that is right. Every side of the big triangle is 3 times the matching side of the small one. In a division the 3 cancels, so the answer is the same.

    2. Not this one. The big triangle has a longer opposite side, 3 times longer. Only the division of two sides stays the same. Go to M4.3.3. Try another.

    3. Not this one. A sum grows when the sides grow. For a triangle 3 times as big, opposite + adjacent is 3 times as big too. Go to M4.3.3. Try another.

    See the answer

    opposite ÷ adjacent

    Every side of the big triangle is 3 times the matching side of the small one. In a division the 3 cancels, so the answer is the same.

    • If you chose B: The big triangle has a longer opposite side, 3 times longer. Only the division of two sides stays the same. Go to M4.3.3.
    • If you chose C: A sum grows when the sides grow. For a triangle 3 times as big, opposite + adjacent is 3 times as big too. Go to M4.3.3.

  6. From an earlier chapter

    The legs of a right angled triangle are 12 cm and 16 cm. How long is the hypotenuse?

    Tap your answer.

    1. Not this one. You added the legs, 12 + 16. The hypotenuse is shorter than that. Square, add, then take the square root. Go to M4.3.1. Try another.

    2. Not this one. You took 12 from 16. For a hypotenuse you add the squares of the legs. Go to M4.3.1. Try another.

    3. Yes, that is right. c² = 12² + 16² = 144 + 256 = 400, and √400 = 20.

    4. Not this one. 400 is c², the hypotenuse squared: 144 + 256. Take the square root to get c. Go to M4.3.1. Try another.

    See the answer

    20 cm

    c² = 12² + 16² = 144 + 256 = 400, and √400 = 20.

    • If you chose A: You added the legs, 12 + 16. The hypotenuse is shorter than that. Square, add, then take the square root. Go to M4.3.1.
    • If you chose B: You took 12 from 16. For a hypotenuse you add the squares of the legs. Go to M4.3.1.
    • If you chose D: 400 is c², the hypotenuse squared: 144 + 256. Take the square root to get c. Go to M4.3.1.

  7. A small triangle has opposite 4 cm and adjacent 5 cm for θ. A bigger one has the same θ and adjacent 35 cm. How many centimetres long is its opposite side?

    See the answer

    28

    The big triangle is 7 times as big, so its opposite is 4 × 7 = 28.

  8. A right angled triangle has opposite 15 cm and adjacent 12 cm for θ. What is opposite ÷ adjacent, as a decimal?

    See the answer

    1.25

    15 ÷ 12 = 1.25.

Keep it

Copy this into your exercise book.

Hypotenuse faces the right angle. Opposite faces θ. Adjacent touches θ. For one angle, opposite ÷ adjacent is the same in every size of triangle.

Draw: A right angled triangle with θ marked, the three sides named, and a bigger copy inside it with the same θ.

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

No internet tomorrow? Save the whole book as one file (776 KB). It opens on any phone.