Lesson 3 ~25 min Unit 3 · Trigonometry +85 XP

Finding a Shorter Side

When the hypotenuse and one leg are known, rearrange $a^2 + b^2 = c^2$ to solve for the missing leg. Subtract from the hypotenuse squared.

Today's hook: A 65-inch TV measures 56.7 inches across. Pythagoras says the height isn't 65 minus 56.7 — it's something smaller. How do you find the missing leg?

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Think First

warm-up

You know the hypotenuse is 13 cm and one leg is 5 cm. Without calculating, do you ADD the squares or SUBTRACT them to find the other leg? Why?

Record your answer in your workbook.

The Big Idea

+5 XP

When the hypotenuse is known, rearrange Pythagoras to: $a^2 = c^2 - b^2$. The shorter side is always found by SUBTRACTING the leg-squared from the hypotenuse-squared.

Start with $a^2 + b^2 = c^2$. To isolate $a^2$, subtract $b^2$ from both sides: $a^2 = c^2 - b^2$. Then square-root. The order matters: the hypotenuse squared comes first, then subtract the known leg squared. Doing it the other way gives a negative number — impossible.

$a = \sqrt{c^2 - b^2}$

Subtract, don't add

Finding a LEG uses minus; finding the hypotenuse uses plus.

Hyp first

$c^2 - b^2$, not $b^2 - c^2$. The hypotenuse-squared is the biggest.

Smaller answer

A leg is always shorter than the hypotenuse — check your answer.

What You'll Master

objectives

Know

The rearranged formula $a^2 = c^2 - b^2$
The hypotenuse squared comes first
A leg is always shorter than the hypotenuse

Understand

Why the hypotenuse squared is the largest of the three
Why $c^2 - b^2$ (not $b^2 - c^2$) gives a positive result
How to identify which side is missing

Can Do

Rearrange $a^2 + b^2 = c^2$ to solve for either leg
Substitute correctly with $c$ as the largest
Sanity-check that the answer is smaller than the hypotenuse

Words You Need

vocabulary

RearrangeUse inverse operations to isolate the unknown letter on one side of the equation.

Inverse operationThe operation that “undoes” another. Subtraction undoes addition; square root undoes squaring.

Leg (shorter side)Either of the two sides that meet at the right angle. Always shorter than the hypotenuse.

Subtraction order$c^2 - b^2$ is positive when $c > b$. The hypotenuse is largest, so this order is correct.

Negative under root$\sqrt{\text{negative}}$ is not a real number. If you get one, you have your $c$ and $b$ swapped.

$c^2$ priorityThe hypotenuse squared is the biggest of the three squares. Always.

Spot the Trap

heads-up

Wrong: “$a^2 = b^2 - c^2$.” Wrong order — this gives a negative number.

Right: $a^2 = c^2 - b^2$ — hypotenuse squared minus leg squared.

Wrong: “$a^2 + b^2 = c^2$, so $a = c - b$.” You can't square-root each term separately.

Right: Subtract the squares, THEN take the square root: $a = \sqrt{c^2 - b^2}$.

Decision Question: Add or Subtract?

+5 XP

Ask yourself: is the unknown the longest side? If yes, ADD ($c^2 = a^2 + b^2$). If no, SUBTRACT ($a^2 = c^2 - b^2$).

Identify what's missing FIRST. If the hypotenuse is missing, ADD the squares of both legs. If a leg is missing, SUBTRACT the known leg squared from the hypotenuse squared.

Missing hyp? Add. Missing leg? Subtract.

What's unknown?

Always check first whether the missing side is the hypotenuse or a leg.

Hyp = largest

$c$ is always the biggest of the three sides.

Sign test

If the result under $\sqrt{}$ is negative, swap $b$ and $c$.

Four-Step Method for a Leg

+5 XP

Solving for a shorter side follows a clear sequence. Practice it until it's automatic.

Step	Action
1	Label sides — mark hypotenuse as $c$
2	Write $a^2 = c^2 - b^2$ (hyp first)
3	Substitute, compute the subtraction
4	Take $\sqrt{}$, round, add units

Label → rearrange → substitute → root

Mark $c$ clearly

Circle the hypotenuse in your diagram before writing equations.

Brackets again

Type $(c^2 - b^2)$ in brackets before the square root.

Answer < hyp

A leg must be less than the hypotenuse — quick sanity check.

Watch Me Solve It · 3 examples

Watch Me Solve It · Standard 13-5-? triangle

+15 XP per step

PROBLEM

A right triangle has hypotenuse 13 cm and one leg 5 cm. Find the other leg.

1

Identify hyp

$c=13$, $b=5$, $a=?$

13 is the biggest, so it's the hypotenuse.
2

Rearrange + substitute

$a^2 = 13^2 - 5^2 = 169 - 25 = 144$
3

Square root

$a = \sqrt{144} = 12$ cm

Recognise 5-12-13 triple.

Answer$a = 12$ cm

Watch Me Solve It · TV diagonal → height

+15 XP per step

PROBLEM

A 65-inch TV has a width of 56.7 inches. Find its height (2 d.p.).

1

Set up

$c = 65$ (diagonal), $b = 56.7$ (width), $a = $ height
2

Compute

$a^2 = 65^2 - 56.7^2 = 4225 - 3214.89 = 1010.11$
3

Root

$a = \sqrt{1010.11} \approx 31.78$ inches

Answer$\approx 31.78$ in

Watch Me Solve It · A non-perfect square

+15 XP per step

PROBLEM

Find the unknown leg of a right triangle with hypotenuse 10 cm and other leg 7 cm. Give the answer to 2 d.p.

1

Rearrange

$a^2 = c^2 - b^2 = 10^2 - 7^2$
2

Compute

$= 100 - 49 = 51$
3

Root

$a = \sqrt{51} \approx 7.14$ cm

Answer$a \approx 7.14$ cm

Common Pitfalls

heads-up

Wrong subtraction order

Computing $b^2 - c^2$ instead of $c^2 - b^2$.

Fix: The hypotenuse squared is always the BIGGER number, so it goes first.

Adding when you should subtract

Treating a leg problem like a hypotenuse problem.

Fix: Ask: is the unknown the longest side? If no → subtract.

Forgetting to identify $c$ first

Substituting the wrong number as the hypotenuse.

Fix: Mark the right angle and find the side opposite it — that's $c$.

Copy Into Your Books

Leg formula

$a^2 = c^2 - b^2$
$a = \sqrt{c^2 - b^2}$
Hyp-squared FIRST

Identify

Find right angle
Opposite side = $c$
$c$ goes into formula first

Method

Label
Rearrange
Substitute
Root + round

Sanity

Leg < hyp
If $\sqrt{}$ of negative, swap
Recognise triples for speed

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Shorter-Side Sprint

4 problems

Four quick drills to lock in today's skill. Try each, then reveal the answer.

1 $c=10$, $b=6$ — find $a$.

$a^2=100-36=64$.$a=8$
2 $c=25$, $b=7$ — find $a$.

$a^2=625-49=576$, $\sqrt{576}=24$.$a=24$
3 $c=17$, $b=15$ — find $a$.

$a^2=289-225=64$.$a=8$
4 $c=20$, $b=12$ — find $a$.

$a^2=400-144=256$, $\sqrt{256}=16$.$a=16$

Complete in your workbook.

Quick Check · 5 questions

Hypotenuse 13, one leg 5. The other leg is:

+10 XP

To find a shorter side you should:

+10 XP

Hyp 25 cm, leg 7 cm. Other leg?

+10 XP

If $\sqrt{c^2 - b^2}$ gives a negative number under the root, what went wrong?

+10 XP

Hyp 10, leg 7. Other leg to 2 d.p.?

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

ApplyMedium3 MARKS

Q6. Find the missing leg in each triangle (2 d.p.): (a) $c=17$, $b=8$ (b) $c=15$, $b=9$ (c) $c=20$, $b=11$

Answer in your workbook.

ApplyEasy2 MARKS

Q7. A 5 m ladder leans against a wall. The base is 1.4 m from the wall. How high up the wall does the top of the ladder reach (2 d.p.)?

Answer in your workbook.

ReasonHard4 MARKS

Q8. A 65-inch TV has aspect ratio 16:9. The diagonal is 65 inches. By letting the width be $16k$ and height $9k$, find $k$ and hence the width and height of the TV (2 d.p.).

Answer in your workbook.

Comprehensive Answers

Quick Check

1. D — 5-12-13 triple.

2. B — $a^2 = c^2 - b^2$.

3. C — 7-24-25 triple.

4. A — $c$ must be the hypotenuse (largest), so $c^2 > b^2$.

5. C — $\sqrt{51}\approx 7.14$.

Show Your Working Model Answers

Q6 (3 marks): (a) $a^2=289-64=225$, $a=15$ [1]. (b) $a^2=225-81=144$, $a=12$ [1]. (c) $a^2=400-121=279$, $a\approx 16.70$ [1].

Q7 (2 marks): Height$^2 = 5^2 - 1.4^2 = 25 - 1.96 = 23.04$ [1]. Height $= \sqrt{23.04} = 4.80$ m [1].

Q8 (4 marks): $(16k)^2 + (9k)^2 = 65^2$ [1]. $256k^2 + 81k^2 = 337k^2 = 4225$ [1]. $k^2 = 12.539...$, $k\approx 3.5410$ [1]. Width $\approx 56.66$ in, height $\approx 31.87$ in [1].

Stretch Challenge · +25 XP, +10 coins

Sliding ladder

A 5 m ladder is set up with its base 3 m from a wall. The top of the ladder reaches some height $h$ up the wall. Someone then pulls the base out so it is 4 m from the wall instead. By how many metres does the top of the ladder drop?

Reveal solution

Original height $= \sqrt{25-9} = 4$ m. New height $= \sqrt{25-16} = 3$ m. The top drops by $4 - 3 = 1$ m.

Quick Review

Formula

$a = \sqrt{c^2 - b^2}$

Order

Hyp squared FIRST, then subtract

Sign

Result under $\sqrt{}$ must be positive

Sanity

Leg shorter than hyp

Identify

Hyp = side opposite right angle

Triples

5-12-13, 7-24-25, 8-15-17, 9-40-41

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