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 measure if both figures must be similar?

On the blueprint of a housing development, the scale used is $1:500$. If the community swimming pool measures $3 \text{ in}$ long on the blueprint, what is its real length expressed in feet?

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 scale factor $k$ and explain how many times greater 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 same instant, the shadow of a nearby utility pole measures $12 \text{ m}$. Applying Thales's theorem, calculate the height of the pole.

A tablet has a rectangular screen whose dimensions are $24 \text{ cm}$ base and $7 \text{ cm}$ height. Calculate the measure of its diagonal applying Pythagoras' theorem.

The roof of a garden shed has the shape of 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.

To secure a radio antenna, a guy wire has been placed from its highest part to the ground. If the antenna is $12 \text{ m}$ high and the wire is tied to the ground at a distance of $5 \text{ m}$ from the base, what is the length of 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 intersect at their midpoint, forming four equal right-angled triangles).

A group of 9th grade students wants to calculate the height of the basketball hoop on the court without using a ladder. At a certain moment, Louis, who is $1,60 \text{ m}$ tall, casts a shadow of $0,80 \text{ m}$. If at that same instant the shadow of the hoop measures $1,52 \text{ m}$, what is the height of the hoop? Draw a diagram of the situation identifying the similar triangles.

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

1. Calculate the real dimensions of the desk in feet.

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

Emma has bought a $50$-inch television. Knowing that $1$ inch is equivalent to $2,54 \text{ in}$, the diagonal of the screen measures exactly $127 \text{ in}$. If the width of the television is $110 \text{ in}$, what is the height of the screen? Use Pythagoras' theorem and round the result to one decimal place.

A cell tower $12 \text{ m}$ high is secured to the ground with two guy wires starting 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 place another similar tower whose height is $18 \text{ m}$, 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 that you must first calculate the apothem using Pythagoras' theorem in one of the equilateral triangles that form the hexagon.

A carpenter is designing a shelving unit that has the shape of a right-angled triangle. He has placed an intermediate shelf parallel to the base. The total height of the shelving unit 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 slanted batten that forms the hypotenuse of the complete shelving unit?

A scale model of a monument has been made using a scale of $1:250$. If the real 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.

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 arrival point is on another platform at a height of $3 \text{ m}$. What is the horizontal distance separating both structures?

A gamer wants to store his mechanical keyboard, which is $52\text{ cm}$ long, inside a carrying case he has bought. The box 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 he places it joining two opposite vertices (the space diagonal)? Calculate the measure of the longest internal diagonal of the box to justify your answer.

A courier company uses drones to deliver packages. For a safe landing on a flat roof, the drone projects two laser beams that form a right angle at the emitter. The drone is at a height $h$ above the roof. The impact points of the lasers on the ground are aligned with the vertical of the drone and are at a distance of $15,6\text{ m}$ from each other. If one of the impact points is $3,9\text{ m}$ from the vertical of 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 real pyramid has a base side of $24\text{ ft}$ and a height of $16\text{ ft}$. If the model has been built with a scale factor of $k = 1:80$, calculate:

1. a) The length of the lateral edge of the real pyramid (you will need to apply Pythagoras' theorem twice).
2. b) The volume of the model in cubic inches, 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 traffic sign has the shape of an equilateral triangle with a side of $90\text{ cm}$. Inside it, there is a red stripe that forms another similar equilateral triangle. If the area of the white part (the inner triangle) is exactly half the total area of the sign, what is the scale factor between the small and the large triangle? What is the side length of the inner triangle? Round to two decimal places.

In a plaza 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 who is $1,65\text{ m}$ tall 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 teenager's head coincides exactly with the top of the flagpole.
2. b) If the teenager walks $2\text{ m}$ towards the flagpole, what length will his shadow projected on the flagpole itself be?

The logo of a new clothing brand is a rhombus inscribed inside a rectangle $16\text{ cm}$ 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 logo similar to the previous one whose perimeter is triple. Justify how the area changes without needing to calculate the new sides.

A lighthouse $35\text{ m}$ tall 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 begins on the cliff) is $100\text{ m}$ and the distance to the highest part 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 pitched roof. The roof has an inclination such that for every $4\text{ m}$ of horizontal advance it rises $3\text{ m}$ in height.

1. a) Calculate the real length of the roof section that covers $8\text{ m}$ of horizontal distance.
2. b) If the solar panel is $2\text{ m}$ long and is placed following the slope of the roof, what vertical height $h$ does the panel cover between its lower and upper edge? (Use similarity of triangles).