Module 14 of 30Lesson 1 of 2 · T-F1 The Right-Triangle Toolkit
Trigonometry strand · Unit T-F1 · Builds towards C6.1

The Right-Triangle Toolkit

🟢 Foundation levelName the sides before you fight

1 Learn it
🔑 In every right-angled triangle: the HYPOTENUSE is the longest side, always OPPOSITE the right angle. Relative to a chosen angle θ: the OPPOSITE side faces θ; the ADJACENT side touches θ (and the right angle). Change the angle, and opposite and adjacent SWAP — the hypotenuse never moves.
θ hypotenuse opposite adjacent
Sides named FROM the angle θ: opposite faces it, adjacent touches it, hypotenuse faces the right angle.

Squares and square roots (N-C3) power everything here: 3² + 4² = 25 = 5². Triples worth recognising instantly: 3-4-5, 5-12-13, 6-8-10, 8-15-17.

2 Watch it work
EXAMPLE 1

A right triangle has sides 3, 4, 5. Which is the hypotenuse?

5 — the longest, opposite the right angle.

EXAMPLE 2

In the figure, θ sits at the bottom-left. Which side is opposite θ?

The vertical side (teal) — it FACES the angle without touching it.

EXAMPLE 3

Check: is a 6-8-10 triangle right-angled?

6² + 8² = 36 + 64 = 100 = 10² ✔ — yes (a scaled 3-4-5).

EXAMPLE 4

Is a 5-6-8 triangle right-angled?

25 + 36 = 61 ≠ 64 — no. The squares must match exactly.

3 Your turn — try first, then reveal
🚫 No calculator[1 mark]

Q1. Which side of a right triangle is always the hypotenuse?

👀 Show answer
✍️ How your working should look in the exam
the longest side, opposite the right angle   ✓ B1
Why this works
Two descriptions, one side — either earns the mark.

🚫 No calculator[1 mark]

Q2. Relative to angle θ, the side that FACES θ is called…

👀 Show answer
✍️ How your working should look in the exam
the opposite   ✓ B1
Why this works
Faces = opposite; touches = adjacent.

🚫 No calculator[2 marks]

Q3. Show whether a 9-12-15 triangle is right-angled.

👀 Show answer
✍️ How your working should look in the exam
81 + 144 = 225 = 15²   ✓ M1
yes — right-angled   ✓ A1
Why this works
A tripled 3-4-5. The squares test settles it beyond doubt.

🚫 No calculator[2 marks]

Q4. Is a 4-7-9 triangle right-angled? Show your check.

👀 Show answer
✍️ How your working should look in the exam
16 + 49 = 65 ≠ 81   ✓ M1
no   ✓ A1
Why this works
The two smaller squares must add EXACTLY to the largest.

🚫 No calculator[1 mark]

Q5. Complete the triple: 5, 12, …

👀 Show answer
✍️ How your working should look in the exam
13   ✓ B1
Why this works
25 + 144 = 169 = 13². Memorised triples save exam minutes.

4 Common mistakes — don’t lose easy marks
  • Opposite/adjacent named from the wrong angle: always name FROM the angle in play.
  • Hypotenuse mislabelled: it faces the RIGHT angle, not the biggest acute angle.
  • Triple-check skipped: 61 ≈ 64 is NOT a right angle — exact or nothing.
  • Adding sides instead of squares in the check.
5 Where this leads

Pythagoras (T-C1) and SOHCAHTOA (T-C2, T-C3) both depend on naming the sides correctly — most trigonometry errors are labelling errors in disguise.

✅ Take the Check Yourself quiz below, then continue to T-C1 Pythagoras.

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