Pythagoras' Theorem
You already know Pythagoras' theorem from earlier years. How long a ladder does a fire crew in Cork need to reach a window $4\text{ m}$ up if the engine has to park $3\text{ m}$ from the wall? In 3rd Year you are going to use Pythagoras' theorem in a much more versatile way: to find the apothem of a hexagon, the diagonal of a whole room, or to set up equations when several pieces of data are missing at once.
The theorem, once more but in greater depth
Before you calculate, estimate. The hypotenuse is always longer than each leg but shorter than the sum of the two legs. This gives you a quick check on your answer.
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.
On a topographic map from Ordnance Survey Ireland (OSi) at a $1:25\,000$ scale, the straight-line distance between the Glen Car Park and the Summit Hut is…
On a topographic map from Ordnance Survey Ireland (OSi) at a $1:25\,000$ scale, the straight-line distance between the Glen Car Park and the Summit Hut is $7.6\text{ cm}$. The contour lines indicate that the station is at an altitude of $1860\text{ m}$ and the hut at $2260\text{ m}$.
- a) Model the terrain profile using a right triangle, indicating what the legs and the hypotenuse represent in physical reality.
- b) Calculate the actual geometric distance in a straight line separating both points across the hillside, expressing the result in meters rounded to the nearest unit.
- c) If a hiker maintains a constant average ascent pace of $1.8\text{ km/h}$ along that straight line, calculate how many minutes it will take to complete the climb.
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-angled triangle hidden 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 calculate the side of a rhombus-shaped reflective traffic sign whose diagonals measure $16\text{ cm}$ and $12\text{ cm}$, Cian writes the following in his notebook:…
To calculate the side of a rhombus-shaped reflective traffic sign whose diagonals measure $16\text{ cm}$ and $12\text{ cm}$, Cian writes the following in his notebook:
$$\ell = \sqrt{16^2 + 12^2} = \sqrt{256 + 144} = \sqrt{400} = 20\text{ cm}$$- a) Identify and explain in detail what conceptual error Cian made when applying the Pythagorean theorem directly to the measurements of the diagonals.
- b) Correct the procedure by justifying which right-angled triangle must be considered and calculate the actual measurement of the side of the sign.
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 slanting side of an isosceles trapezium are all calculated in the same way: by drawing the height, which always falls perpendicular and forms a right-angled 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 centre to that chord always form a right-angled triangle: this is the relationship that solves problems about pipes, channels 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 centre 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 centre 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 obtain a diagonal of that face, and then by combining that diagonal with the height to obtain the space diagonal.
To transport a rigid metal bar, a cuboid-shaped box (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 cuboid-shaped box (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.
The Great Pyramid of Giza has a square base. Suppose the side of its base measures $230\text{ m}$ and the length of one of its side edges (from a base vertex to the…
The Great Pyramid of Giza has a square base. Suppose the side of its base measures $230\text{ m}$ and the length of one of its side edges (from a base vertex to the apex) is $219\text{ m}$. Calculate the height $h$ of the pyramid.
A room is in the shape of a cuboid 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…
A room is in the shape of a cuboid 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 has to be solved.
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.
The urban planning department of a city council is going to delineate a landscaped flowerbed in the shape of a right triangle. It has been set as a condition that the…
The urban planning department of a city council is going to delineate a landscaped flowerbed in the shape of a right triangle. It has been set as a condition that the total perimeter of the flowerbed must be exactly $60\text{ m}$ and that the hypotenuse measures $26\text{ m}$.
- a) Set up a quadratic equation that allows you to find the length of the legs from the given data, clearly defining what the unknown $x$ represents.
- b) Solve the equation you set up to determine the exact dimensions of both legs.
- c) Justify algebraically whether it would be possible to design another flowerbed in the shape of a right triangle with that same hypotenuse of $26\text{ m}$ but with a total perimeter of $68\text{ m}$.
The altitude to the hypotenuse
Drawing the altitude from the right angle to the hypotenuse splits the triangle into two smaller right-angled triangles. Each has the same angles as the whole triangle, so all three are similar [Theorem 13]. Let the altitude be $h$ and let it divide the hypotenuse $a$ into $p$ and $q$.
In the design of an industrial warehouse roof, the main frame forms a right-angled triangle. The altitude to the hypotenuse divides it into two segments of $4\text{ m}$…
In the design of an industrial warehouse roof, the main frame forms a right-angled triangle. The altitude to the hypotenuse divides it into two segments of $4\text{ m}$ and $9\text{ m}$. Apply the altitude theorem to calculate the length of this altitude.
In a park, a flower bed is in the shape of a right-angled triangle. The hypotenuse measures $25\text{ m}$ and the projection of one of the legs onto the hypotenuse…
In a park, a flower bed is in the shape of a right-angled triangle. The hypotenuse measures $25\text{ m}$ and the projection of one of the legs onto the hypotenuse measures $9\text{ m}$. Use the leg theorem to calculate the length of that leg.
In a right-angled triangle, the altitude to the hypotenuse measures $12\text{ cm}$ and one of the legs measures $15\text{ cm}$. Using the leg theorem and altitude…
In a right-angled triangle, the altitude to the hypotenuse measures $12\text{ cm}$ and one of the legs measures $15\text{ cm}$. Using the leg theorem and altitude theorem, or Pythagoras' Theorem iteratively, calculate the length of the hypotenuse $a$ and the other leg $c$.