Exercises: Applications of Pythagoras' theorem

A graphic designer has a rectangular photograph $10 \text{ cm}$ wide and $15 \text{ cm}$ high. If they decide to enlarge it to make a poster so that the new width is $40 \text{ cm}$, what will the height of the poster be if both shapes must be similar?

On the map of a housing estate, the scale used is $1:500$. If the communal swimming pool measures $3 \text{ cm}$ long on the map, what is its actual length in metres?

Two squares are similar. The side of the first one measures $4 \text{ cm}$ and the side of the second one measures $12 \text{ cm}$. Calculate the similarity ratio $k$ and explain how many times larger the area of the second square is compared to the first.

At a certain time of day, a vertical post $2 \text{ m}$ high casts a shadow of $1,5 \text{ m}$. At that exact same moment, the shadow of a nearby electricity pylon measures $12 \text{ m}$. Applying Thales' theorem, calculate the height of the pylon.

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.

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.

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?

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).

Marco has to calculate the side length of a rhombus-shaped reflective sign whose diagonals measure $16\text{ cm}$ and $12\text{ cm}$. To solve it, he writes in his notebook: $l^2 = 16^2 + 12^2 = 256 + 144 = 400$, concluding that the side length is $l = \sqrt{400} = 20\text{ cm}$. Identify the geometric error Marco made, explain why the Pythagorean theorem cannot be applied directly with those data, and calculate the correct side length of the rhombus.

A patio tile is shaped like a regular hexagon with a side length of $10\text{ cm}$. Remember that in a regular hexagon, the radius is exactly equal to the side length.

1. a) Before performing any numerical calculations, reason whether the apothem of the tile must be greater than, less than, or equal to $10\text{ cm}$, and estimate whether its value will be closer to $5\text{ cm}$ or $9\text{ cm}$. Justify your estimate by considering the relationship between the legs and the hypotenuse.
2. b) Set up the corresponding right triangle, calculate the exact length of the apothem rounded to two decimal places, and check whether your initial estimate was correct.

A group of 2nd ESO students wants to calculate the height of the basketball hoop in the playground without using a ladder. At a given moment, Luis, who is $1,60 \text{ m}$ tall, casts a shadow of $0,80 \text{ m}$. If at that exact same moment the basketball hoop's shadow is $1,52 \text{ m}$ long, what is the height of the basketball hoop? Draw a diagram of the situation identifying the similar triangles.

You can do it on paper and send a photo with your answer.

On a floor plan of a room drawn to a scale of $1:50$, the dimensions of a rectangular desk are $2,4 \text{ cm}$ wide and $3,6 \text{ cm}$ long.

1. Calculate the actual dimensions of the desk in metres.

2. If the actual area of the room is $12 \text{ m}^2$, what is the area occupied by the room on the floor plan expressed in $\text{cm}^2$?

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.

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.

1. What is the total length of wire needed to secure the mast?

2. 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?

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.

A carpenter is designing a bookcase shaped like a right-angled triangle. He has placed an intermediate shelf parallel to the base. The total height of the bookcase is $150 \text{ cm}$ and the base measures $90 \text{ cm}$. If the intermediate shelf is located at a height of $60 \text{ cm}$ from the base:

1. What is the length of this intermediate shelf?

2. What is the length of the sloped wooden beam forming the hypotenuse of the full bookcase?

A scale model of a monument has been made using a scale of $1:250$. If the actual monument has a volume of $1500 \text{ m}^3$, calculate the volume of the model in cubic decimetres ($\text{dm}^3$). Remember the relationship between the scale factor and the ratio of the volumes.

To cross a small ravine in an adventure park, a zip line $25 \text{ m}$ long has been installed. The starting point is on a tower at a height of $18 \text{ m}$ and the end point is on another platform at a height of $3 \text{ m}$. What is the horizontal distance separating both structures?

At an AEMET weather station in the Sierra de Guadarrama, a vertical wind mast $12\text{ m}$ high is going to be installed. To ensure its stability against wind gusts, it will be held by two guy wires anchored from the top end of the mast to two anchor points on the horizontal ground. Due to the rocky terrain, the first anchor is placed $5\text{ m}$ from the base of the mast, and the second anchor, on the opposite side, is located $9\text{ m}$ from that base. Knowing that each wire requires an additional $25\text{ cm}$ at each of its two ends to make the regulatory ties, calculate how many metres of wire must be purchased in total for the two guy wires. Round the final result to two decimal places.

A road signage company commissions the design of a rhombus-shaped warning sign whose outer side must measure exactly $25\text{ cm}$.

1. a) Write the equation that relates the length of the major diagonal ($D$), the minor diagonal ($d$), and the side of the rhombus using the Pythagorean theorem.
2. b) Following the reverse approach, propose two different combinations of positive integers for the full diagonal measurements ($D$ and $d$) that result in an exact side length of $25\text{ cm}$. Mathematically justify with the relevant calculations why both proposals are valid.

A gamer wants to fit their $52\text{ cm}$ long mechanical keyboard inside a carrying case they bought. The case is a cuboid with internal dimensions of $48\text{ cm}$ long, $18\text{ cm}$ wide and $10\text{ cm}$ high. Will the keyboard fit in the box if placed connecting two opposite vertices (the space diagonal)? Calculate the length of the space diagonal inside the box to justify your answer.

A courier company uses drones to deliver packages. To ensure a safe landing on a rooftop, the drone projects two laser beams that form a right angle at the emitter. The drone is at a height $h$ above the rooftop. The laser impact points on the ground are aligned with the vertical below the drone and are $15,6\text{ m}$ apart. If one of the impact points is $3,9\text{ m}$ from the vertical below the drone, calculate:

1. a) The height $h$ at which the drone is flying using the altitude theorem.
2. b) The distance from the drone to each of the two landing points (the legs of the triangle).

For a film festival, a scale model of a regular square-based pyramid has been built. The actual pyramid has a base side length of $24\text{ m}$ and a height of $16\text{ m}$. If the model was built with a scale factor of $k = 1:80$, calculate:

1. a) The length of the slant edge of the actual pyramid (you will need to apply Pythagoras' theorem twice).
2. b) The volume of the model in cubic centimetres, knowing that the volume of a pyramid is $V = \frac{1}{3} \cdot \text{Base area} \cdot \text{Height}$. (Use the relationship $k^3$ for the volume).

A road sign is in the shape of an equilateral triangle with a side length of $90\text{ cm}$. Inside it, a red border forms another similar equilateral triangle. If the area of the white section (the inner triangle) is exactly half of the total area of the sign, what is the scale factor between the smaller and larger triangles? How long is the side of the inner triangle? Round to two decimal places.

In a town square, there is a vertical flagpole. At a distance of $12\text{ m}$ from its base, there is a spotlight on the ground projecting light. A teenager measuring $1,65\text{ m}$ in height stands between the spotlight and the flagpole, at a distance of $3\text{ m}$ from the spotlight.

1. a) Calculate the height of the flagpole knowing that the shadow of the boy's head coincides exactly with the top of the flagpole.
2. b) If the boy walks $2\text{ m}$ towards the flagpole, how long will his shadow cast on the flagpole itself be?

The logo for a new clothing brand is a rhombus inscribed inside a rectangle measuring $16\text{ cm}$ long and $12\text{ cm}$ wide. The vertices of the rhombus are the midpoints of the sides of the rectangle. Calculate:

1. a) The perimeter of the rhombus.
2. b) The area of a second similar logo whose perimeter is triple the first. Justify how the area changes without calculating the new side lengths.

A lighthouse $35\text{ m}$ high is situated on a vertical cliff. From a boat at sea, it is observed that the distance to the base of the lighthouse (the point where the lighthouse meets the cliff) is $100\text{ m}$ and the distance to the top of the lighthouse is $110\text{ m}$. Calculate:

1. a) The height of the cliff above sea level.
2. b) The horizontal distance from the boat to the base of the cliff.

A rectangular solar panel is to be installed on a sloping roof. The roof has a pitch such that for every $4\text{ m}$ of horizontal distance, it rises by $3\text{ m}$ in height.

1. a) Calculate the actual length of the section of roof that covers a horizontal distance of $8\text{ m}$.
2. b) If the solar panel is $2\text{ m}$ long and is placed along the slope of the roof, what vertical height $h$ does the panel span between its bottom and top edges? (Use similarity of triangles).

A homeowners' association in Valencia is going to install a continuous straight ramp to replace an outdoor staircase made up of $4$ identical steps. Each step has a tread (horizontal depth) of $18\text{ cm}$ and a riser (vertical height) of $15\text{ cm}$. To order the inclined handrail that will run along the entire ramp, they need to know its total length. Two board members propose different methods:

• Sonia's method: Calculate the hypotenuse of a single step by applying Pythagoras to its measurements ($18\text{ cm}$ and $15\text{ cm}$) and multiply the result obtained by $4$.

• Roberto's method: First add the measurements of the $4$ steps to find the total base ($4 \cdot 18 = 72\text{ cm}$) and the total height ($4 \cdot 15 = 60\text{ cm}$), and apply Pythagoras once to the overall right triangle.

1. a) Work through both methods showing the calculations to two decimal places and check whether the same total length is obtained.
2. b) Prove, using triangle similarity, why both procedures are mathematically equivalent.
3. c) Justify which of the two methods is more advisable if the goal is to minimise the accumulation of rounding errors when the step measurements have inexact decimals.

A single-family home in Segovia is going to renovate its symmetrical gable roof (with an isosceles triangle profile) to install photovoltaic solar panels. The horizontal facade of the house is $8\text{ m}$ wide, and the building has a depth of $12\text{ m}$. To comply with technical regulations regarding snow loads and allow for proper water drainage, the vertical height of the ridge (the top vertex of the roof) must be at least $3\text{ m}$, but it cannot exceed $4,5\text{ m}$ due to municipal planning restrictions. The architect has two types of wooden beams available for the sloping roof pitch:

• Beam A: $4,5\text{ m}$ in length.

• Beam B: $5,5\text{ m}$ in length.

1. a) Identify the right triangle of the roof profile, calculate to two decimal places the ridge height that would be achieved with each type of beam, and deduce which of the two options is the only one that complies with the technical regulations.
2. b) For the regulatory beam option, calculate the total area of the sloping roof by adding the two rectangular slopes of the house.
3. c) The solar panels are rectangular and measure $2\text{ m}$ long by $1\text{ m}$ wide, and they will be installed only on the south-facing slope (a rectangle $12\text{ m}$ deep by the length of the valid beam). Analyse how it is best to orient the panels (horizontally or vertically relative to the ground) so as not to exceed the edges of the roof, and calculate the maximum number of complete panels that can be placed.