Exercises: Pythagoras' Theorem

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 $12\text{ m}$. 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.

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 side of the rhombus).

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.

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 measures $9\text{ m}$. Use the leg theorem to calculate the length of that leg.

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:

1. a) Identify and explain in detail what conceptual error Cian made when applying the Pythagorean theorem directly to the measurements of the diagonals.
2. b) Correct the procedure by justifying which right-angled triangle must be considered and calculate the actual measurement of the side of the sign.

You want to check whether a rectangular plywood board can pass diagonally through the doorway of a classroom at your school that measures $2.1\text{ m}$ in height by $0.8\text{ m}$ in width.

1. a) Without performing the square root calculation, make a reasoned estimation: explain why the diagonal of the opening must necessarily be greater than $2.1\text{ m}$ and less than $2.9\text{ m}$, and propose an approximate value to one decimal place, justifying your deduction.
2. b) Calculate the exact length of the diagonal by applying the Pythagorean theorem (rounding to two decimal places) and check whether your previous estimate was consistent with the established bounds.

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.

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 $30\text{ m}$ 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 centre of the pipe, what is the maximum depth of the water $h$?

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.

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.

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 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.

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 $2260\text{ m}$.

1. a) Model the terrain profile using a right triangle, indicating what the legs and the hypotenuse represent in physical reality.
2. 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.
3. 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.

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 $26\text{ m}$.

1. 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.
2. b) Solve the equation you set up to determine the exact dimensions of both legs.
3. 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 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.

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.

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 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).

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 $c$.

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.

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.

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?

A vertical telecommunications tower $24\text{ m}$ high is secured to the ground by two taut cables anchored at points $A$ and $B$, located on opposite sides relative to the base of the tower $P$ and aligned with it. The cable anchored at $A$ measures $25\text{ m}$ and the one anchored at $B$ measures $30\text{ m}$. You want to calculate the total ground distance between both anchor points, $d(A,B)$.

Two students propose different methods:

• Cian states: «I can directly consider the large triangle formed by the two cables and the ground distance, so $d(A,B) = \sqrt{25^2 + 30^2}$».

• Aoife states: «That method is not valid; you must apply the Pythagorean theorem to each side of the tower separately and add the legs along the base: $d(A,B) = \sqrt{25^2 - 24^2} + \sqrt{30^2 - 24^2}$».

1. a) Evaluate the validity of Cian's proposal, explaining why it fails from the perspective of the geometric hypotheses of the Pythagorean theorem.
2. b) Work through Aoife's method to calculate the distance $d(A,B)$ exactly and to one decimal place.
3. c) Imagine a different hypothetical case in which the two cables formed a right angle ($90^\circ$) with each other at the top of the tower. In that scenario, which of the two strategies would be more direct to find $d(A,B)$? Justify your answer by comparing both approaches.

A special heavy-goods transport vehicle must pass through a one-way mountain tunnel whose cross-section is a perfect semicircle of radius $R = 5\text{ m}$. The roadway occupies the horizontal diameter of the semicircle. The vehicle carries a container with a rectangular cross-section of total width $3.6\text{ m}$. Safety regulations require the vehicle to travel centered on the roadway and to maintain a vertical safety clearance of at least $0.4\text{ m}$ between the top corner of the container and the tunnel vault.

1. a) Model the cross-section of the tunnel using a geometric system, identifying the right-angled triangle that relates the radius of the tunnel, the half-width of the container, and the maximum height of the ceiling above its corner.
2. b) Apply the Pythagorean theorem to deduce the maximum total height the container can have to legally travel centered through the tunnel, rounding to two decimal places.
3. c) Due to roadworks on the roadway, the truck is diverted to the right, such that the right side of the container is only $0.5\text{ m}$ from the outer edge of the tunnel roadway. Reason geometrically whether the allowable height will increase or decrease compared to the centered case, and calculate the new maximum allowed height analytically.