Master the art of choosing the right tool. Learn when to reach for Pythagoras and when to use trigonometry, then combine both to crack multi-step problems.
Today's hook: A carpenter has a right-angled triangular brace with two sides known: 5 cm and 12 cm. They need the third side. Then the foreman asks for the angle opposite the 5 cm side. Two different tools for two different questions -- and the second answer depends on the first.
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A right-angled triangle has two sides given: 5 cm and 12 cm. Would you use Pythagoras or trigonometry to find the third side? What if you were given one side and one angle instead?
Record your answer in your workbook.
1
The Big Idea
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Pythagoras needs two sides and finds the third. Trigonometry needs one angle and one side, then finds anything else. The key is knowing which tool matches your given information -- and recognising when a problem needs both, one after the other.
An isosceles right triangle has two equal sides and two 45° angles. Its hypotenuse is always the leg times $\sqrt{2}$. This gives exact trig values for 45° without a calculator: $\sin 45° = \cos 45° = \frac{1}{\sqrt{2}}$ and $\tan 45° = 1$.
$a^2 + b^2 = c^2$ vs $\sin \theta = \frac{O}{H}$
Two sides = Pythagoras
If no angles are given, trig ratios have nothing to pair with.
Angle + side = trig
Once you have an angle, SOH CAH TOA unlocks everything.
Both = multi-step
Pythagoras first to find a side, then trig to find the angle.
2
What You'll Master
objectives
Know
When Pythagoras is appropriate and when trigonometry is appropriate
That isosceles right triangles have angles of 45°, 45°, 90°
The exact ratios for 45° derived from an isosceles right triangle
Understand
Why some problems require both Pythagoras and trigonometry in sequence
That finding a missing side can be a stepping stone to finding a missing angle
Can Do
Decide whether a problem needs Pythagoras, trigonometry, or both
Solve multi-step problems by combining the two techniques
Work with isosceles right triangles efficiently
3
Words You Need
vocabulary
Pythagoras' Theorem$a^2 + b^2 = c^2$, used when two sides of a right-angled triangle are known and a third side is required.
Trigonometric RatiosSOH CAH TOA, used when one angle (other than 90°) and one side are known.
Isosceles Right TriangleA right-angled triangle with two equal sides and two 45° angles.
Multi-step ProblemA problem requiring more than one calculation or technique to reach the final answer.
Exact ValueAn answer expressed as a surd or fraction, not a rounded decimal.
HypotenuseThe longest side of a right-angled triangle, opposite the right angle.
4
Spot the Trap
heads-up
Wrong: Using Pythagoras when an angle is given. If you know an angle and a side, you must use trigonometry.
Right: Ask "What do I know, and what do I want to find?" Two sides only → Pythagoras. One angle + one side → Trigonometry.
Wrong: Rounding intermediate answers too early. This introduces errors in the second step.
Right: Store exact values in your calculator, or keep surd form, until the final step. Round only at the end.
5
Parts of the Whole
+5 XP
Every right-angled triangle problem contains the same three ingredients: what you know, what you want, and the relationship that connects them. The relationship is either Pythagoras (sides only) or trigonometry (angles and sides).
Before calculating, scan the given information. Two sides and no angles -- Pythagoras. One angle and one side -- trigonometry. Two sides and you need an angle -- Pythagoras first, then inverse trig.
$c = \sqrt{a^2 + b^2}$
Scan before calculating
Identify what is given before choosing a formula.
No angle = no trig
Trig ratios need an angle. Without one, use Pythagoras.
Recognise Pythagorean triples
3-4-5, 5-12-13, 8-15-17 save time and verify answers.
6
Pick the Tool
+5 XP
The decision is binary. Two sides given? Pythagoras. One angle and one side given? Trigonometry. The only trick is recognising when a problem disguises its given information inside a word description or diagram.
Pythagoras connects three sides. Trigonometry connects two sides and an angle. You cannot use trig without an angle (other than 90°), and you cannot use Pythagoras to find an angle. Each tool has one job.
$a^2 + b^2 = c^2$ | $\tan \theta = \frac{O}{A}$
Pythagoras = sides only
If the problem mentions no angles, Pythagoras is your only move.
Trig = angle + side
Once an angle is involved, SOH CAH TOA takes over.
Both = sequential
Pythagoras first finds the missing side. Then trig finds the angle.
7
Two-Step Problems
+5 XP
Many problems require you to find one unknown before you can tackle another. Treat each step as a separate right-angled triangle problem. The answer from step one becomes the given information for step two.
Step 1: Identify what you can find immediately. Step 2: Use that new value as given information. Step 3: State the final answer with correct units and rounding. Always draw a fresh diagram and re-label for each step.
$c = 10$ → $\sin \theta = \frac{6}{10}$
Keep exact values
Do not round intermediate answers. Store them in your calculator.
Re-label each step
Redraw and re-label to avoid mixing up opposite and adjacent.
Round at the end only
Use full precision until the very last line of working.
8
Exact Values
+5 XP
An isosceles right triangle has two equal sides and two 45° angles. These triangles generate exact trigonometric values for 45° -- no calculator needed.
If the equal sides have length $a$, the hypotenuse is $a\sqrt{2}$. This means $\sin 45° = \cos 45° = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$ and $\tan 45° = 1$. Memorise these -- they appear constantly.
$\sin 45° = \frac{\sqrt{2}}{2}$, $\tan 45° = 1$
Memorise the three values
sin 45° = cos 45° = 1/√2. tan 45° = 1. Always.
Rationalise if needed
1/√2 = √2/2. Both are correct but √2/2 is standard.
Square diagonals use this
A square's diagonal splits it into two isosceles right triangles.
Watch Me Solve It · 3 examples
Watch Me Solve It · Choosing the tool
+15 XP per step
Q1
PROBLEM
A right-angled triangle has hypotenuse 13 cm and one other side 5 cm. Find the remaining side.
1
Identify the given information
Hypotenuse = 13 cm, one leg = 5 cm, no angles given
Two sides known, no angle mentioned. This decides the tool.
The exact value matches what we memorised for 45°.
4
Approximate check
$7\sqrt{2} \approx 7 \times 1.414 \approx 9.90$ cm
The hypotenuse should be longer than either leg. 9.90 > 7 ✓
Nice work -- XP earned
AnswerHypotenuse = $7\sqrt{2}$ cm ≈ 9.90 cm, angles = 45° each
9
Common Pitfalls
heads-up
Using Pythagoras when an angle is given
If the problem gives you an angle and a side, Pythagoras alone cannot find anything. You need trigonometry because the angle creates a ratio between sides.
Fix: scan for angles first. If any angle other than 90° is mentioned, trig is involved.
Rounding intermediate answers too early
Finding a hypotenuse as 9.9 cm, then using 9.9 in the next step instead of the exact value. This introduces rounding errors that compound.
Fix: store intermediate values in your calculator memory, or keep them in surd form, until the final answer.
Forgetting that isosceles right triangles have 45° angles
Trying to calculate the angles of an isosceles right triangle using inverse trig when they are obviously 45° each. This wastes time and invites calculator errors.
Fix: if two sides are equal and there is a right angle, the other two angles are 45°. No calculation needed.
Copy Into Your Books
Decision Framework
Two sides only → Pythagoras
One angle + one side → SOH CAH TOA
Two sides + need angle → both
Isosceles Right Triangle
Angles: 45°, 45°, 90°
Hypotenuse = leg × √2
Leg = hypotenuse / √2
Exact Values for 45°
sin 45° = 1/√2 = √2/2
cos 45° = 1/√2 = √2/2
tan 45° = 1
Multi-Step Strategy
Step 1: find what you can
Step 2: use as new given
Step 3: round only at the end
How are you completing this lesson?
Brain Trainer · 4 problems
D
Brain Trainer · Mixed
4 problems
Four problems combining Pythagoras and trigonometry. Work each one, then reveal the answer.
1 A right-angled triangle has legs 9 cm and 12 cm. Find the hypotenuse.
In a right-angled triangle, you know two side lengths but no angles. Which tool should you use?
Correct -- two sides and no angles means Pythagoras.
Not quite -- trig ratios need an angle to pair with a side.
Explanation: Pythagoras' theorem ($a^2 + b^2 = c^2$) connects three sides. Trigonometry needs an angle (other than 90°) to create a ratio. With only sides given, Pythagoras is the only option.
MC2
Classic Pythagorean triple
+10 XP
A right-angled triangle has sides 5 cm and 12 cm meeting at the right angle. What is the length of the hypotenuse?
Correct -- $5^2 + 12^2 = 25 + 144 = 169 = 13^2$.
Check your arithmetic -- this is the classic 5-12-13 triangle.
Explanation: $c = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$ cm. The 5-12-13 triangle is one of the most common Pythagorean triples.
MC3
Isosceles right triangle hypotenuse
+10 XP
In an isosceles right triangle, each equal side is 4 cm. The hypotenuse is:
Correct -- hypotenuse = leg × √2 = $4\sqrt{2}$ cm.
In an isosceles right triangle, the hypotenuse is the leg multiplied by √2, not 2 or √3.
Sine = opposite / hypotenuse. The opposite side is 6 and the hypotenuse is 10.
Explanation: $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{6}{10} = \frac{3}{5} = 0.6$. No need to find the other leg first -- sine only needs opposite and hypotenuse.
MC5
Two-step sequence
+10 XP
A right-angled triangle has legs 3 cm and 4 cm. You need to find the angle opposite the 3 cm side. What is the correct sequence?
Correct -- find hypotenuse (5) first, then use sin−1(3/5).
You cannot use inverse trig without knowing the hypotenuse.
Explanation: First find the hypotenuse using Pythagoras: $\sqrt{3^2 + 4^2} = 5$ cm. Then $\sin \theta = \frac{3}{5}$, so $\theta = \sin^{-1}(0.6) \approx 36.9°$. Option A uses tangent correctly but the question asks for the angle opposite 3, and tan uses opposite/adjacent which works too -- but the standard approach taught is to find the hypotenuse first.
Short Answer · 3 questions
Q6
Two-step triangle
+15 XP
Q6
SHORT ANSWER
A right-angled triangle has legs of length 9 cm and 12 cm. (a) Find the length of the hypotenuse. (b) Find the size of the angle opposite the 9 cm side, correct to one decimal place. (c) Verify that the other acute angle is 90° minus your answer from (b).
(c) The other acute angle = $90° - 36.9° = 53.1°$. Check: $\sin 53.1° \approx 0.8 = \frac{12}{15}$. Verified.
Q7
Isosceles right triangle exact values
+15 XP
Q7
SHORT ANSWER
An isosceles right triangle has equal sides of length $x$ cm. (a) Show that the hypotenuse has length $x\sqrt{2}$ cm. (b) Write exact values for $\sin 45°$, $\cos 45°$ and $\tan 45°$ using this triangle. (c) A square has diagonals of length 16 cm. By considering the diagonal as the hypotenuse of an isosceles right triangle formed by two adjacent sides, find the side length of the square.
A ramp in a Sydney train station rises 1.5 metres over a horizontal distance of 4 metres. A right-angled triangle models this ramp with the vertical rise as one leg and the horizontal run as the other leg. (a) Find the length of the ramp (the hypotenuse), correct to two decimal places. (b) Find the angle the ramp makes with the horizontal, correct to one decimal place. (c) If the ramp must not exceed an angle of 22° with the horizontal for accessibility, does this ramp meet the requirement? Justify with a calculation. (d) A second ramp has the same rise but is twice as long horizontally. Without calculating, explain whether its angle with the horizontal will be greater or less than the first ramp.
(c) Yes. $20.6° < 22°$, so the ramp meets the accessibility requirement.
(d)Less than. With the same rise but a longer horizontal run, the ramp is shallower. Since $\tan \theta = \frac{\text{rise}}{\text{run}}$, doubling the run halves the tangent value, so the angle decreases.
An equilateral triangle has side length 10 cm. A perpendicular is drawn from one vertex to the opposite side, splitting the triangle into two right-angled triangles. (a) Find the exact height of the equilateral triangle. (b) Using one of these right-angled triangles, write exact values for $\sin 60°$, $\cos 60°$ and $\tan 60°$.
Record in your book -- full marks require clear working.
The perpendicular bisects the base, creating two right triangles with hypotenuse 10 cm and base 5 cm.
Use the interactive below to practise deciding whether to use Pythagoras or trigonometry, then solve step-by-step. Try to make the decision before the interactive reveals it.
Record two observations about how you decide which tool to use.
Consolidation Game -- Doodle Jump Quiz
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Jump your way to the top by answering questions on Pythagoras, trigonometry and combined problems. The higher you climb, the harder the questions.
Lesson Complete
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Mark this lesson as complete to earn your bonus XP.