Applications of the Pythagorean 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 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.

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?

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?

The roof of a garden shed is shaped like an isosceles triangle. The base of the triangle measures $6 \text{ meters}$ and the equal sides measure $5 \text{ meters}$ each. Calculate the height of the roof.

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' theorem in one of the equilateral triangles that form the hexagon.

A gamer wants to fit their $52\text{ cm}$ long mechanical keyboard inside a carrying case they bought. The case is a rectangular prism 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 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 development, the scale used is $1:500$. If the community swimming pool measures $3 \text{ cm}$ long on the map, what is its actual length in meters?

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 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.

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?

A cell phone tower $12 \text{ m}$ 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 tower.

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

2. If we want to set up another similar tower with a height of $18 \text{ m}$, at what distance from the base should we anchor the wires so that the structure remains proportional?

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 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).

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.

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 meters.

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 $\text{cm}^2$?

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 decimeters ($\text{dm}^3$). Remember the relationship between the scale factor and the ratio of the volumes.

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 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 the Pythagorean theorem twice).
2. b) The volume of the model in cubic centimeters, knowing that the volume of a pyramid is $V = \frac{1}{3} \times \text{Base area} \times \text{Height}$. (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.

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.