Applications of 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.

Aoife 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 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.

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?

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.

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.

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.

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).

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.

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.

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?

A group of 2nd Year students wants to calculate the height of the basketball hoop in the playground without using a ladder. At a given moment, Seán, 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.

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?