Before you start, make sure you know: The Pythagorean Theorem (Secondary Year 2 (first cycle))
Before you start, make sure you know: Quadratic Equations (Secondary Year 3 (first cycle))
A builder wants the corner of a house on a lot to be perfectly square. He marks $3\text{ m}$ on one wall, $4\text{ m}$ on the other, and adjusts until the string between the two marks measures exactly $5\text{ m}$. Why does that measurement guarantee a right angle? In this section you will prove it, and you will also use the theorem to calculate diagonals, heights, apothems, chords, and distances in space.
The Theorem and Its Proof
Change the values and see how the area of the square on the hypotenuse is always the sum of the areas of the squares on the legs. But seeing is not proving. Let's look at a proof using areas.
In the section on the altitude and the leg you will see a second proof, using similar triangles.
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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.
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.
Inside the diagonal of a rectangle, the diagonal of a square, or the side of a rhombus there is always a hidden right triangle. You have to find it.
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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.
An isosceles triangle is not a right triangle, but its altitude cuts it into two identical right triangles, because it falls on the midpoint of the base.
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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.
The cross-section of an irrigation canal is in the shape of an isosceles trapezium. The width at the top is $8\text{ m}$, the width at the bottom is $2\text{ m}$ and the…
The cross-section of an irrigation canal is in the shape of an isosceles trapezium. The width at the top is $8\text{ m}$, the width at the bottom is $2\text{ m}$ and the depth of the canal is $4\text{ m}$. Calculate the length of the sloping sides $x$ of the canal.
In a trapezoid, when you draw the altitudes from the endpoints of the shorter base, right triangles appear on the sides. Always ask yourself: what part of the base falls in each triangle?
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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.
A vertical aerial is held by two cables anchored to the ground on opposite sides of the base. One cable measures $15\text{ m}$ and is anchored $9\text{ m}$ from the…
A vertical aerial is held by two cables anchored to the ground on opposite sides of the base. One cable measures $15\text{ m}$ and is anchored $9\text{ m}$ from the base. The other cable is anchored $16\text{ m}$ from the base. Calculate the height of the aerial and the length of the second cable.
The altitude to the hypotenuse divides the right triangle into two smaller triangles. All three triangles are similar (they have the same angles), and very useful relationships come from this.
Here $a$ is the hypotenuse, $b$ and $c$ are the legs, $m$ is the projection of $b$ onto the hypotenuse, and $n$ is the projection of $c$.
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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.
The roof of a house is in the shape of a right-angled triangle resting on its hypotenuse. The height of the roof relative to the horizontal base is $6\text{ m}$ and the…
The roof of a house is in the shape of a right-angled triangle resting on its hypotenuse. The height of the roof relative to the horizontal base is $6\text{ m}$ and the projection of one of the roof slopes (leg) onto the base is $8\text{ m}$. Calculate the length of the other projection and the lengths of the two sloping sides of the roof.
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 $c$.
The radius, half of a chord, and the distance from the center to the chord form a right triangle. This is so because the radius perpendicular to a chord bisects it.
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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.
In three dimensions, Pythagoras is applied twice in a row: first on a face to find a flat diagonal, and then with the height to find the space diagonal.
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To transport a rigid metal bar, a rectangular prism-shaped box with dimensions $80\text{ cm}$ long, $60\text{ cm}$ wide, and $50\text{ cm}$ high is available. Calculate…
To transport a rigid metal bar, a rectangular prism-shaped box 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 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).
Sometimes you don't know any complete side, only relationships between them. Then you call the unknown $x$ and the theorem gives you an equation (or a system).
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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?
Go back to the beginning: the $5\text{ m}$ string guarantees the square corner because $3^2 + 4^2 = 5^2$, and by the converse of the theorem that triangle has a right angle.
Let's practice! 8 medium exercises in this section