Exercises: Pythagoras' theorem

A fire truck ladder leans against the facade 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 facade. Calculate the total length of the ladder.

A telecommunications mast is supported by a taut cable 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 at what distance 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 out the actual size of the screen.

A town square is shaped like a square with a side of $50\text{ m}$. 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 kite in the shape of a rhombus, 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).

A traffic sign is in the shape of an isosceles triangle. Its base measures $60\text{ cm}$ and each of the two equal sides measures $50\text{ cm}$. Calculate the height $h$ of the sign.

In the design of the roof of an industrial building, the main structure forms a right 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 flowerbed is in the shape of a right 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 cross-section of an irrigation canal is in the shape of an isosceles trapezoid. 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.

A decorative mosaic is made up of tiles in the shape of a regular hexagon with a side of $10\text{ cm}$. Calculate the apothem of one of these tiles.

A plot of land is in the shape of a right trapezoid. The side perpendicular to the 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 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 $h$?

The roof of a house is in the shape of a right triangle resting on its hypotenuse. The height of the roof with respect to the horizontal base is $6\text{ m}$ and the projection of one of the slopes (leg) onto the base is $8\text{ m}$. Calculate the length of the other projection and the measure of the two sloping sides of the roof.

To transport a rigid metal bar, a box in the shape of a rectangular prism is available, whose dimensions are $80\text{ cm}$ long, $60\text{ cm}$ wide and $50\text{ cm}$ high. Calculate the maximum length the bar can have to fit completely enclosed inside the box.

A vertical antenna is supported 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 antenna and the length of the second cable.

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.

The Great Pyramid of Giza has a square base. Suppose that the side of its base measures $230\text{ m}$ and that the length of one of its lateral edges (from a vertex of the base to the apex) is $219\text{ m}$. Calculate the height $h$ of the pyramid.

In a garden, an area in the shape of a right triangle is to be marked out. It is known that the lengths of its three sides are three consecutive even integers. Set up the corresponding equation using the Pythagorean theorem and determine the lengths of the three sides.

In a circular piece of wood with a radius of $25\text{ cm}$, two straight and parallel cuts are made. One of the cuts 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 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 of the floor by walking along the walls, ceiling or floor. If the spider takes the shortest possible path, calculate this distance. (Hint: imagine the net of the room).

In a right triangle, the altitude to the hypotenuse measures $12\text{ cm}$ and one of the legs measures $15\text{ cm}$. Using the leg and altitude theorems, or the Pythagorean theorem iteratively, calculate the length of the hypotenuse $a$ and the other leg $c$.

A bridge has a lateral structure in the shape of a scalene trapezoid. The horizontal bases measure $40\text{ m}$ and $26\text{ m}$, and the oblique sides measure $13\text{ m}$ and $15\text{ m}$. By setting up a system of equations with the Pythagorean theorem, calculate the height of the trapezoid.

A logo is formed by an isosceles triangle inscribed in a circle. The base of the triangle measures $16\text{ cm}$ and its altitude to the base measures $18\text{ cm}$. Calculate the radius of the circle in which the triangle is inscribed.

Two towers $15\text{ m}$ and $20\text{ m}$ high 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, covering exactly $25\text{ m}$. Calculate the distance between the bases of the towers. Next, 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 cover?