Applications of Pythagoras' theorem
Pythagoras' theorem almost never appears on its own: you have to find the right-angled triangle hidden in a screen, a roof or a guy wire, and then work out whatever you are asked for.
The diagonal of a rectangle: screens and photos
In any rectangle, the diagonal, the width and the height form a right-angled triangle: the diagonal is the hypotenuse.
A tablet has a rectangular screen with dimensions of $24 \text{ cm}$ base and $7 \text{ cm}$ height. Calculate the length of its diagonal applying Pythagoras' theorem.
A tablet has a rectangular screen with dimensions of $24 \text{ cm}$ base and $7 \text{ cm}$ height. Calculate the length of its diagonal applying Pythagoras' theorem.
Marta has bought a $50$ inch television. Knowing that $1$ inch is equivalent to $2,54 \text{ cm}$, the diagonal of the screen measures exactly $127 \text{ cm}$. If the…
Marta has bought a $50$ inch television. Knowing that $1$ inch is equivalent to $2,54 \text{ cm}$, the diagonal of the screen measures exactly $127 \text{ cm}$. If the width of the television is $110 \text{ cm}$, what is the height of the screen? Use Pythagoras' theorem and round your answer to one decimal place.
Guy wires and aerials
A guy wire holding up a vertical pole, the pole itself and the stretch of ground between them also form a right-angled triangle: the wire is the hypotenuse.
To secure a radio mast, a guy wire has been attached from its top to the ground. If the mast is $12 \text{ m}$ high and the wire is anchored to the ground at a distance…
To secure a radio mast, a guy wire has been attached from its top to the ground. If the mast is $12 \text{ m}$ high and the wire is anchored to the ground at a distance of $5 \text{ m}$ from the base, how long is the wire?
A mobile phone mast $12 \text{ m}$ high is anchored to the ground with two guy wires extending from its top end. Each wire is anchored at a distance of $5 \text{ m}$…
A mobile phone mast $12 \text{ m}$ high is anchored to the ground with two guy wires extending from its top end. Each wire is anchored at a distance of $5 \text{ m}$ from the base of the mast.
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What is the total length of wire needed to secure the mast?
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If we want to set up another similar mast with a height of $18 \text{ m}$, at what distance from the base should we anchor the wires so that the structure remains proportional?
Roofs and isosceles triangles
A pitched roof has the shape of an isosceles triangle. Its height, half of the base and one of its sides form a right-angled triangle: draw the height and you have the usual triangle.
The roof of a garden shed is shaped like an isosceles triangle. The base of the triangle measures $6 \text{ metres}$ and the equal sides measure $5 \text{ metres}$ each.…
The roof of a garden shed is shaped like an isosceles triangle. The base of the triangle measures $6 \text{ metres}$ and the equal sides measure $5 \text{ metres}$ each. Calculate the height of the roof.
Polygons with hidden right-angled triangles: rhombuses and hexagons
The diagonals of a rhombus intersect at right angles and at their midpoints: they divide the rhombus into four equal right-angled triangles. And in a regular hexagon, the apothem, the side and half of the side form another right-angled triangle.
Calculate the perimeter of a rhombus whose diagonals measure $16 \text{ cm}$ and $12 \text{ cm}$. (Hint: Remember that the diagonals of a rhombus are perpendicular and…
Calculate the perimeter of a rhombus whose diagonals measure $16 \text{ cm}$ and $12 \text{ cm}$. (Hint: Remember that the diagonals of a rhombus are perpendicular and bisect each other, forming four congruent right-angled triangles).
Calculate the perimeter and the area of a regular hexagon whose side measures $8 \text{ cm}$. To find the area, remember to first calculate the apothem using Pythagoras'…
Calculate the perimeter and the area of a regular hexagon whose side measures $8 \text{ cm}$. To find the area, remember to first calculate the apothem using Pythagoras' theorem in one of the equilateral triangles that form the hexagon.