If you lean a ladder against a wall, the ladder, the wall and the ground form a right-angled triangle. More than $2\,000$ years ago, Pythagoras discovered a fixed relationship between the three sides of any right-angled triangle: if you know two sides, you can always calculate the third.
Before you start, make sure you know: Triangles (1st ESO)
Before you start, make sure you know: Square root (1st ESO)
Pythagoras' theorem
In a right-angled triangle, the two sides that form the right angle are called the legs ($a$ and $b$); the opposite side, the longest one, is the hypotenuse ($c$).
Look at the drawing: the square built on the hypotenuse has exactly the same area as the two squares on the legs put together. That is why the formula works.
Practice★☆☆Not started
In a football training session, a player runs in a straight line from one corner of the pitch to the opposite corner. If the pitch is $80$ metres long and $60$ metres…
In a football training session, a player runs in a straight line from one corner of the pitch to the opposite corner. If the pitch is $80$ metres long and $60$ metres wide, what distance does the player cover in their diagonal run?
If you know the two legs, solve directly for $c$ from the formula.
Practice★☆☆Not started
We want to attach a cable from the top of a $12$ metre high mobile phone mast to an anchor point on the ground, $9$ metres from the base of the mast. How long must the…
We want to attach a cable from the top of a $12$ metre high mobile phone mast to an anchor point on the ground, $9$ metres from the base of the mast. How long must the cable be so that it is pulled completely taut?
To secure a volleyball net in the school playground, a guy wire has been attached from the top of a post to an anchor on the ground. The post has a height of $2,4$…
To secure a volleyball net in the school playground, a guy wire has been attached from the top of a post to an anchor on the ground. The post has a height of $2,4$ metres and the distance from the base of the post to the anchor is $1,8$ metres. If we want to attach an identical wire to another post that is a scaled enlargement by a factor of $k = 1,5$ of the first one, how long will the new guy wire be?
If you know the hypotenuse and one leg, solve for the missing leg by subtracting instead of adding.
Practice★☆☆Not started
Television screens are measured by their diagonal in inches. If a television has a width of $32$ cm and a height of $24$ cm, how many centimetres is its diagonal?
Television screens are measured by their diagonal in inches. If a television has a width of $32$ cm and a height of $24$ cm, how many centimetres is its diagonal?
In a park, there is a skate ramp whose side view forms a right-angled triangle. The base of the ramp measures $4\text{ metres}$ and its maximum height is…
In a park, there is a skate ramp whose side view forms a right-angled triangle. The base of the ramp measures $4\text{ metres}$ and its maximum height is $3\text{ metres}$. At a distance of $1,6\text{ metres}$ from the base, a vertical support beam has been placed for reinforcement.
a) Calculate the total length of the ramp (the sloped part).
b) Using similar triangles, calculate the height of the vertical support beam.
Pythagoras also works in three dimensions. To find the diagonal of a box (a cuboid), you apply the theorem twice: first on the base, then with the height.
Practice★★★Not started
A courier service uses cardboard boxes in the shape of a cuboid (rectangular prism). One of its largest boxes has dimensions of $60\text{ cm}$ long, $45\text{ cm}$ wide…
A courier service uses cardboard boxes in the shape of a cuboid (rectangular prism). One of its largest boxes has dimensions of $60\text{ cm}$ long, $45\text{ cm}$ wide and $30\text{ cm}$ high. A customer needs to send a rigid tube measuring exactly $82\text{ cm}$.
Use Pythagoras' theorem to determine whether the tube fits inside the box (placing it diagonally connecting two opposite vertices of the internal space) and justify your answer with the necessary calculations.
Back to the start: If you lean a $5$ m ladder so that its foot is $3$ m from the wall, the height it reaches is the missing leg: $h = \sqrt{5^2 - 3^2} = \sqrt{25 - 9} = \sqrt{16} = 4$ m.