UNIT 9 • GEOMETRY & TRIGONOMETRY

Pythagoras Theorem
Find the Missing Side

The Pythagoras theorem says that in a right-angled triangle $a^2 + b^2 = c^2$ , where $c$ is the hypotenuse (the longest side, opposite the right angle). Know any two sides and you can find the third.

HOW IT WORKSRIGHT TRIANGLES ONLY

\begin{gathered} c^2 = a^2 + b^2 \\ c = \sqrt{a^2 + b^2} \\ a = \sqrt{c^2 - b^2} \end{gathered}

1. Find the HypotenuseIt is opposite the right angle and is always the longest side,

c

2. Add or SubtractFinding

c

: add the squares. Finding a leg: subtract the squares from

c^2

3. Square RootTake the square root. Leave it as a surd (like

5\sqrt{2}

) if the question asks for an exact answer.

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

A 13 m ladder leans against a wall and reaches a window 12 m above the ground. How far is the foot of the ladder from the wall?

STEP 01 • NAME THE SIDES

The wall and the ground meet at a right angle. The ladder is the hypotenuse $c = 13$ , the height is $b = 12$ , and the distance we want is $a$ .

STEP 02 • SUBTRACT THE SQUARES

a^2 = c^2 - b^2 = 13^2 - 12^2 = 169 - 144 = 25

STEP 03 • SQUARE ROOT

a = \sqrt{25} = 5 \text{ m}

Check: $5^2 + 12^2 = 25 + 144 = 169 = 13^2$ ✓ (5-12-13 is a Pythagorean triple).

FINAL CONCLUSION

The foot of the ladder is 5 m from the wall

a = 5 \text{ m}

The calculator below opens on this ladder and also gives the angle it makes with the ground.

LABORATORY

Interactive Pythagoras Calculator

LABORATORY

Pythagoras Engine

Enter any two sides and leave the third empty. Enter all three to check whether the triangle is right-angled.

Leg aLeg bHypotenuse c (longest side)

Common Questions

What is the Pythagoras theorem?

In a right-angled triangle, the square on the hypotenuse equals the sum of the squares on the other two sides:

a^2 + b^2 = c^2

, where

c

is the side opposite the right angle.

How do you find a shorter side?

Rearrange:

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

. Subtract, don't add, when the hypotenuse is the side you know.

What are Pythagorean triples?

Whole numbers that fit

a^2 + b^2 = c^2

: 3-4-5, 5-12-13, 8-15-17, 7-24-25, and any multiple of them such as 6-8-10.

How do you check if a triangle is right-angled?

Square the two shorter sides and add them. If the total equals the square of the longest side, the triangle has a right angle (the converse of Pythagoras). If it is smaller, the largest angle is obtuse; if bigger, all angles are acute.

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Unit 9•41 of 47

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f(x)∫dxπ

NETZ Engine

Evaluating numerical algorithm

UNIT 9 • GEOMETRY & TRIGONOMETRY

Pythagoras Theorem
Find the Missing Side

HOW IT WORKSRIGHT TRIANGLES ONLY

\begin{gathered} c^2 = a^2 + b^2 \\ c = \sqrt{a^2 + b^2} \\ a = \sqrt{c^2 - b^2} \end{gathered}

1. Find the HypotenuseIt is opposite the right angle and is always the longest side,

c

2. Add or SubtractFinding

c

: add the squares. Finding a leg: subtract the squares from

c^2

3. Square RootTake the square root. Leave it as a surd (like

5\sqrt{2}

) if the question asks for an exact answer.

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

A 13 m ladder leans against a wall and reaches a window 12 m above the ground. How far is the foot of the ladder from the wall?

STEP 01 • NAME THE SIDES

The wall and the ground meet at a right angle. The ladder is the hypotenuse $c = 13$ , the height is $b = 12$ , and the distance we want is $a$ .

STEP 02 • SUBTRACT THE SQUARES

a^2 = c^2 - b^2 = 13^2 - 12^2 = 169 - 144 = 25

STEP 03 • SQUARE ROOT

a = \sqrt{25} = 5 \text{ m}

Check: $5^2 + 12^2 = 25 + 144 = 169 = 13^2$ ✓ (5-12-13 is a Pythagorean triple).

FINAL CONCLUSION

The foot of the ladder is 5 m from the wall

a = 5 \text{ m}

The calculator below opens on this ladder and also gives the angle it makes with the ground.

LABORATORY

Interactive Pythagoras Calculator

LABORATORY

Pythagoras Engine

Enter any two sides and leave the third empty. Enter all three to check whether the triangle is right-angled.

Leg aLeg bHypotenuse c (longest side)

Common Questions

What is the Pythagoras theorem?

In a right-angled triangle, the square on the hypotenuse equals the sum of the squares on the other two sides:

a^2 + b^2 = c^2

, where

c

is the side opposite the right angle.

How do you find a shorter side?

Rearrange:

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

. Subtract, don't add, when the hypotenuse is the side you know.

What are Pythagorean triples?

Whole numbers that fit

a^2 + b^2 = c^2

: 3-4-5, 5-12-13, 8-15-17, 7-24-25, and any multiple of them such as 6-8-10.

How do you check if a triangle is right-angled?

PreviousVolume & Surface Area

Unit 9•41 of 47

NextTriangle Solver

Pythagoras Theorem Find the Missing Side

Step-by-Step Worked Example

The foot of the ladder is 5 m from the wall

Interactive Pythagoras Calculator

Pythagoras Engine

Common Questions

NETZ Engine

Pythagoras Theorem Find the Missing Side

Step-by-Step Worked Example

The foot of the ladder is 5 m from the wall

Interactive Pythagoras Calculator

Pythagoras Engine

Common Questions

Pythagoras Theorem
Find the Missing Side

Pythagoras Theorem
Find the Missing Side