The Pythagorean Theorem
How long is the diagonal of your cell phone screen? Does the fire department's ladder reach the third-floor window? Both questions are answered with the same result: the Pythagorean theorem. In this section you will see what it says, why it is true, and how to use it to find the apothem of a hexagon, the diagonal of a box, or the sides of a triangle when all you know is how they are related.
The theorem
In a right triangle, the two sides that form the right angle are called the legs, and the side opposite the right angle, the longest one, is the hypotenuse.
A fire engine ladder rests against the façade of a building. The base of the ladder is $3\text{ m}$ from the wall and reaches a height of $4\text{ m}$ on the wall.…
A fire engine ladder rests against the façade of a building. The base of the ladder is $3\text{ m}$ from the wall and reaches a height of $4\text{ m}$ on the wall. Calculate the total length of the ladder.
A telecommunications mast is held by a cable tensioned from its top end to a point on the ground. The cable measures $13\text{ m}$ and the mast has a height of…
A telecommunications mast is held by a cable tensioned from its top end to a point on the ground. The cable measures $13\text{ m}$ and the mast has a height of $12\text{ m}$. Determine how far from the base of the mast the cable is anchored to the ground.
Why is it true?
The fact that $3^2 + 4^2 = 5^2$ works in one example proves nothing: it could be a coincidence. To be sure it holds for all right triangles, we need to justify it. The clearest way is to compare areas.
Take four identical copies of a right triangle with legs $b$ and $c$ and hypotenuse $a$, and arrange them inside a square of side $b + c$, as in the drawing. A gap is left in the middle: a square of side $a$.
The dimensions of a rectangular computer screen are $32\text{ cm}$ wide and $24\text{ cm}$ high. Calculate the length of its diagonal to find the actual screen size.
The dimensions of a rectangular computer screen are $32\text{ cm}$ wide and $24\text{ cm}$ high. Calculate the length of its diagonal to find the actual screen size.
Diagonals and rhombuses
The diagonal of a rectangle, the side of a rhombus from its diagonals, or the diagonal of a square are classic cases: there is always a right triangle hiding between two known measurements and the one you are looking for.
A town square is in the shape of a square with side length $50\text{ m}$. If you decide to cross it in a straight line from one corner to the opposite corner, what…
A town square is in the shape of a square with side length $50\text{ m}$. If you decide to cross it in a straight line from one corner to the opposite corner, what distance will you cover?
To build a rhombus-shaped kite, two wooden rods measuring $60\text{ cm}$ and $80\text{ cm}$ have been crossed. Calculate the length of the outer edge of the kite (the…
To build a rhombus-shaped kite, two wooden rods measuring $60\text{ cm}$ and $80\text{ cm}$ have been crossed. Calculate the length of the outer edge of the kite (the side of the rhombus).
The perimeter of a rhombus-shaped swimming pool is $52\text{ m}$ and one of its diagonals measures $10\text{ m}$. Calculate the length of the other diagonal and the…
The perimeter of a rhombus-shaped swimming pool is $52\text{ m}$ and one of its diagonals measures $10\text{ m}$. Calculate the length of the other diagonal and the surface area of the pool.
Heights and apothems in polygons
The height of an isosceles triangle, the apothem of a regular polygon, or the slanted side of an isosceles trapezoid are all calculated the same way: by drawing the height, which always falls perpendicular and forms a right triangle with half of the base.
A road sign is in the shape of an isosceles triangle. Its base measures $60\text{ cm}$ and each of its two equal sides measures $50\text{ cm}$. Calculate the height $h$…
A road sign is in the shape of an isosceles triangle. Its base measures $60\text{ cm}$ and each of its two equal sides measures $50\text{ cm}$. Calculate the height $h$ of the sign.
A decorative mosaic is made of regular hexagonal tiles with a side length of $10\text{ cm}$. Calculate the apothem of one of these tiles.
A decorative mosaic is made of regular hexagonal tiles with a side length of $10\text{ cm}$. Calculate the apothem of one of these tiles.
A plot of land is in the shape of a right-angled trapezium. The side perpendicular to the parallel bases measures $12\text{ m}$, the longer base measures $30\text{ m}$…
A plot of land is in the shape of a right-angled trapezium. The side perpendicular to the parallel bases measures $12\text{ m}$, the longer base measures $30\text{ m}$ and the shorter base measures $25\text{ m}$. Calculate the total perimeter of the plot.
Chords and circles
The radius of a circle, half of a chord, and the distance from the center to that chord always form a right triangle: this is the relationship that solves problems about pipes, irrigation canals, or circumscribed circles.
A cylindrical pipe carries water. The radius of the pipe is $13\text{ cm}$ and the surface of the water forms a horizontal chord $24\text{ cm}$ wide. If the water level…
A cylindrical pipe carries water. The radius of the pipe is $13\text{ cm}$ and the surface of the water forms a horizontal chord $24\text{ cm}$ wide. If the water level is below the center of the pipe, what is the maximum depth of the water $h$?
In a circular piece of wood with a radius of $25\text{ cm}$, two straight parallel cuts are made. One cut has a length of $48\text{ cm}$ and the other $40\text{ cm}$. If…
In a circular piece of wood with a radius of $25\text{ cm}$, two straight parallel cuts are made. One cut has a length of $48\text{ cm}$ and the other $40\text{ cm}$. If both cuts are made on opposite sides of the center of the circle, calculate the perpendicular distance separating the two cuts.
A logo is formed by an isosceles triangle inscribed in a circle. The base of the triangle measures $16\text{ cm}$ and its height relative to the base measures…
A logo is formed by an isosceles triangle inscribed in a circle. The base of the triangle measures $16\text{ cm}$ and its height relative to the base measures $18\text{ cm}$. Calculate the radius of the circle in which the triangle is inscribed.
Pythagoras in space
In three dimensions, Pythagoras is applied twice in a row: first on a face to get a flat diagonal, and then combining that diagonal with the height to get the space diagonal.
To transport a rigid metal bar, a box shaped like a rectangular prism with dimensions $80\text{ cm}$ long, $60\text{ cm}$ wide and $50\text{ cm}$ high is available.…
To transport a rigid metal bar, a box shaped like a rectangular prism with dimensions $80\text{ cm}$ long, $60\text{ cm}$ wide and $50\text{ cm}$ high is available. Calculate the maximum length the bar can have so that it fits completely inside the closed box.
A room is in the shape of a rectangular prism with dimensions $5\text{ m}$ long, $4\text{ m}$ wide and $3\text{ m}$ high. A spider is in one corner of the ceiling and…
A room is in the shape of a rectangular prism with dimensions $5\text{ m}$ long, $4\text{ m}$ wide and $3\text{ m}$ high. A spider is in one corner of the ceiling and wants to reach the opposite corner on the floor by crawling along the walls, ceiling or floor. If the spider takes the shortest possible path, calculate this distance. (Hint: consider the net of the room).
Pythagoras with equations
When you do not know any side completely, only relationships between them, Pythagoras becomes an equation (or a system) that you have to solve.
In a garden, a section in the shape of a right-angled triangle is to be enclosed. It is known that the lengths of its three sides are three consecutive even integers.…
In a garden, a section in the shape of a right-angled triangle is to be enclosed. It is known that the lengths of its three sides are three consecutive even integers. Set up the corresponding equation using Pythagoras' Theorem and determine the lengths of all three sides.
A bridge has a side frame in the shape of a scalene trapezium. The horizontal parallel bases measure $40\text{ m}$ and $26\text{ m}$, and the non-parallel sides measure…
A bridge has a side frame in the shape of a scalene trapezium. The horizontal parallel bases measure $40\text{ m}$ and $26\text{ m}$, and the non-parallel sides measure $13\text{ m}$ and $15\text{ m}$. Set up a system of equations using Pythagoras' Theorem to calculate the height of the trapezium.
Two towers of height $15\text{ m}$ and $20\text{ m}$ are separated by an unknown horizontal distance. A bird flies in a straight line from the top of the $15\text{ m}$…
Two towers of height $15\text{ m}$ and $20\text{ m}$ are separated by an unknown horizontal distance. A bird flies in a straight line from the top of the $15\text{ m}$ tower to the base of the $20\text{ m}$ tower, travelling exactly $25\text{ m}$. Calculate the distance between the bases of the towers. Then, if another bird flies in a straight line from the top of the $20\text{ m}$ tower to the base of the $15\text{ m}$ tower, what distance will it travel?