Lesson 2 ~25 min Unit 3 · Measurement & Geometry +85 XP

Finding a Shorter Side

Rearrange Pythagoras to find a missing leg: $a = \sqrt{c^2 - b^2}$.

Today's hook: A 10 m ladder leans against a wall, its base 4 m from the wall. How high up the wall does it reach? You know the hypotenuse and one leg — can you find the other?

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Think First

warm-up

In a right-angled triangle, the hypotenuse is 13 cm and one leg is 5 cm. What do you think the other leg is? How could you work it out using what you know about Pythagoras' Theorem?

Record your answer in your workbook.

The Big Idea

+5 XP

When the hypotenuse $c$ and one leg are known, rearrange the theorem to subtract and square-root your way to the missing leg.

Start from $c^2 = a^2 + b^2$. If $c$ and $b$ are known, subtract $b^2$ from both sides: $a^2 = c^2 - b^2$. Then square root: $a = \sqrt{c^2 - b^2}$. The key change is subtraction instead of addition.

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

Find $c$ first

Always identify which side is the hypotenuse before rearranging.

Subtract, not add

Finding a leg means $c^2 - b^2$, not $c^2 + b^2$.

Verify your answer

Check by substituting back: does $a^2 + b^2 = c^2$?

What You'll Master

objectives

Know

The rearranged formula: $a = \sqrt{c^2 - b^2}$
That subtraction replaces addition when finding a leg
How to verify by substituting back

Understand

Why the formula rearranges by subtracting $b^2$
Why the hypotenuse must be identified before rearranging

Can Do

Find a missing leg given the hypotenuse and other leg
Apply this to real-world problems (ladders, tents, wires)
Verify answers by substitution

Words You Need

vocabulary

RearrangeChange the subject of a formula by applying the same operation to both sides.

Subject of a formulaThe variable isolated on one side, e.g. in $a = \sqrt{c^2-b^2}$, $a$ is the subject.

VerifyCheck an answer by substituting it back into the original equation.

Inverse operationThe reverse operation — subtraction undoes addition; square root undoes squaring.

Slant heightThe diagonal length along the surface of a shape (often the hypotenuse in problems).

Perpendicular heightThe vertical measurement at 90° to the base — one of the legs in a right triangle.

Spot the Trap

heads-up

Wrong: $c=10, b=6$, so $a = \sqrt{10^2 + 6^2} = \sqrt{136}$

Right: $a = \sqrt{10^2 - 6^2} = \sqrt{64} = 8$ — subtract, not add!

Wrong: Treating a leg as the hypotenuse: using $c=6$ when the largest side is 10.

Right: Always assign the largest value to $c$ before substituting.

Rearranging the Formula

+5 XP

From $c^2 = a^2 + b^2$, subtract $b^2$ from both sides to isolate $a^2$, then square root.

Start: $\;c^2 = a^2 + b^2$
Subtract $b^2$: $\;c^2 - b^2 = a^2$
Flip sides: $\;a^2 = c^2 - b^2$
Square root: $\;a = \sqrt{c^2 - b^2}$

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

Identifying Which Side Is Missing

+5 XP

Before calculating, label the triangle: find $c$ (opposite right angle), find the known leg, mark the unknown with a question mark.

Three questions to ask:

Which side is opposite the right angle? → That's $c$.
Which side is given? → That's $b$ (or $a$).
Which side is unknown? → That's what we calculate.

If unknown side is longest → use addition ($c = \sqrt{a^2+b^2}$).
If unknown side is a shorter side → use subtraction ($a = \sqrt{c^2-b^2}$).

Missing leg: subtract. Missing hypotenuse: add.

Calculator Technique

+5 XP

To compute $a = \sqrt{c^2 - b^2}$: calculate $c^2 - b^2$ first, then press the square root key.

Example: $c = 10, b = 6$

Key in: $10^2 - 6^2 =$
Display: $64$
Press $\sqrt{\phantom{x}}$
Display: $8$ — answer is $a = 8$

Always compute the subtraction before pressing $\sqrt{}$. Round to 2 decimal places unless told otherwise.

Compute $c^2 - b^2$ first, then press $\sqrt{}$.

Checking Your Answer

+5 XP

Always verify: substitute all three sides into $a^2 + b^2 = c^2$. If both sides match, you're correct.

Example: Found $a = 8$, with $b = 6$, $c = 10$.

Check: $8^2 + 6^2 = 64 + 36 = 100 = 10^2$ ✓

For decimal answers: a small rounding difference (e.g. 99.96 ≈ 100) is acceptable.

Substitute all 3 sides back: $a^2 + b^2 \stackrel{?}{=} c^2$

Common Pitfalls

heads-up

Adding instead of subtracting

Using $c^2 + b^2$ when finding a leg, producing an answer larger than the hypotenuse.

Fix: Finding a shorter side always means subtracting: $c^2 - b^2$.

Mislabelling the hypotenuse

Treating a leg as $c$, causing a negative number under the square root.

Fix: $c$ is always the longest side, opposite the right angle — check first.

Skipping verification

Accepting any decimal without checking it satisfies $a^2 + b^2 = c^2$.

Fix: Always substitute all three values back to confirm.

Copy Into Your Books

Finding a Shorter Side

$a = \sqrt{c^2 - b^2}$
$c$ = hypotenuse (longest)
Subtract the known leg²

Steps

1. Identify $c$ (hypotenuse)
2. Write $a^2 = c^2 - b^2$
3. Calculate $c^2 - b^2$
4. Square root the result

Verify

Substitute all 3 sides back
Check $a^2 + b^2 = c^2$
Small decimal rounding is OK

Key Rule

Finding hypotenuse → ADD
Finding a leg → SUBTRACT
$c$ must be the biggest value!

How are you completing this lesson?

Watch Me Solve It · 3 examples

Watch Me Solve It · Basic Leg

+15 XP per step

PROBLEM

A right-angled triangle has hypotenuse $c = 10$ and leg $b = 6$. Find the other leg $a$.

1
Write the rearranged formula
$a^2 = c^2 - b^2$
Subtract the known leg² from the hypotenuse².
2
Substitute and calculate
$a^2 = 10^2 - 6^2 = 100 - 36 = 64$
3
Square root and verify
$a = \sqrt{64} = 8$
Verify: $8^2 + 6^2 = 100 = 10^2$ ✓

Answer$a = 8$

Watch Me Solve It · 5-12-13 Triple

+15 XP per step

PROBLEM

Hypotenuse $c = 13$, leg $b = 5$. Find $a$.

1
Write the formula
$a^2 = c^2 - b^2$
2
Substitute
$a^2 = 13^2 - 5^2 = 169 - 25 = 144$
3
Square root
$a = \sqrt{144} = 12$
5-12-13 is a core Pythagorean triple.

Answer$a = 12$

Watch Me Solve It · Tent Height

+15 XP per step

PROBLEM

A tent has a slant side (hypotenuse) of 3.5 m and a half-width of 2.1 m. Find the height of the tent.

1
Identify the sides
$c = 3.5$ m (slant side), $b = 2.1$ m (half-width), $a =$ height
The tent cross-section forms a right triangle: slant is the hypotenuse.
2
Substitute
$a^2 = 3.5^2 - 2.1^2 = 12.25 - 4.41 = 7.84$
3
Square root and verify
$a = \sqrt{7.84} = 2.8$ m
Check: $2.8^2 + 2.1^2 = 7.84 + 4.41 = 12.25 = 3.5^2$ ✓

AnswerTent height $= 2.8$ m

Brain Trainer · 4 problems

Brain Trainer · Shorter Side Drills

4 problems

Find the missing leg for each triangle. Work it out, then reveal the answer.

1 $c = 17$, $b = 15$. Find $a$.
$a^2 = 289 - 225 = 64$ → $a = 8$
2 $c = 25$, $b = 7$. Find $a$.
$a^2 = 625 - 49 = 576$ → $a = 24$
3 $c = 15$, $b = 9$. Find $a$.
$a^2 = 225 - 81 = 144$ → $a = 12$
4 $c = 7.5$, $b = 4.5$. Find $a$.
$a^2 = 56.25 - 20.25 = 36$ → $a = 6$

Complete in your workbook.

Quick Check · 5 questions

What is the correct formula to find a leg $a$, given $c$ and $b$?

+10 XP

$c = 17$, $b = 15$. Find $a$.

+10 XP

$c = 25$, $b = 7$. Find $a$.

+10 XP

$c = 15$, $b = 9$. Find $a$.

+10 XP

A 15 m rope runs from the top of a vertical pole to a point 9 m from the base. How tall is the pole?

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

ApplyMedium3 MARKS

Q6. A right-angled triangle has hypotenuse 26 cm and one leg 24 cm. Find the other leg and verify your answer by substitution.

Answer in your workbook.

UnderstandEasy2 MARKS

Q7. A 5.3 m wire runs from the top of a 5 m flag pole to a point on the ground. How far is that point from the base of the pole? Give your answer to 1 decimal place.

Answer in your workbook.

ReasonHard4 MARKS

Q8. Two vertical poles are 4 m apart. Pole A is 3 m tall and Pole B is 8 m tall. A wire runs from the top of Pole A to the base of Pole B, and another from the top of Pole B to the base of Pole A. Find the length of each wire to 2 decimal places.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. A — $a = \sqrt{c^2 - b^2}$ is the correct rearrangement.

2. A — $\sqrt{289-225} = \sqrt{64} = 8$.

3. D — $\sqrt{625-49} = \sqrt{576} = 24$.

4. C — $\sqrt{225-81} = \sqrt{144} = 12$.

5. B — height $= \sqrt{225-81} = 12$ m.

Model Answers

Q6 (3 marks): $a^2 = 26^2 - 24^2 = 676 - 576 = 100$. $a = 10$ cm. Verify: $10^2 + 24^2 = 676 = 26^2$ ✓.

Q7 (2 marks): $d^2 = 5.3^2 - 5^2 = 28.09 - 25 = 3.09$. $d = \sqrt{3.09} \approx 1.8$ m.

Q8 (4 marks): Wire A-top to B-base: legs 3 m and 4 m, length $= \sqrt{9+16} = 5$ m. Wire B-top to A-base: legs 8 m and 4 m, length $= \sqrt{64+16} = \sqrt{80} \approx 8.94$ m.

Stretch Challenge · +25 XP, +10 coins

Double Pole Wire Problem

Two poles are 10 m apart. One is 8 m tall and the other is 14 m tall. A wire connects the top of each pole to the top of the other. Find the length of this diagonal wire. Show all working.

Reveal solution

Vertical difference between tops: $14 - 8 = 6$ m. Horizontal distance: 10 m. Wire length $= \sqrt{6^2 + 10^2} = \sqrt{136} \approx 11.66$ m.

Quick Review

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

Subtract the known leg squared from the hypotenuse squared, then square root.

Identify $c$ first

The hypotenuse is always the largest side, opposite the right angle.

Subtract for a leg

Finding a shorter side means subtraction. Finding the hypotenuse means addition.

Calculator steps

Compute $c^2 - b^2$ first, then press the $\sqrt{}$ key.

Always verify

Substitute all three values back: $a^2 + b^2$ must equal $c^2$.

No negatives under root

If you get a negative, you've swapped a leg and hypotenuse — correct and retry.

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