Applications of the Pythagorean theorem
Wilkin wants to know if his 6-inch cell phone fits in a pocket, and Yaritza needs to know how much cable it takes to hold up an antenna. In both cases there is a right triangle hidden. They almost never give it to you drawn: you have to find it, and then apply the Pythagorean theorem or similarity. At the end we will come back to Wilkin's phone.
The diagonal of a rectangle: screens and photos
In any rectangle, the diagonal, the width, and the height form a right triangle. The diagonal is the hypotenuse. Remember: $c^2 = a^2 + b^2$.
If you are given the diagonal and one side, the other side is a leg: you subtract squares, $a^2 = c^2 - b^2$.
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.
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…
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.
Cables, ramps, and zip lines
A guy wire, the vertical pole, and the ground form a right triangle. The cable is the hypotenuse because it is the longest side and is opposite the right angle.
The same idea works for a zip line or a ramp. If the starting point is at $18$ m and the landing point at $3$ m, the drop in height is $18 - 3 = 15$ m: that is one leg. The horizontal distance is the other leg, and the zip line is the hypotenuse.
If one object sits on top of another (a lighthouse on a cliff), draw one triangle for each line of sight. You will see that the horizontal distance to the foot of the wall is shared by both triangles.
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…
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…
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…
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:
- a) The height of the cliff above sea level.
- b) The horizontal distance from the boat to the base of the cliff.
Roofs and isosceles triangles
A gable roof has the shape of an isosceles triangle. If you draw the altitude, it falls at the midpoint of the base and you get two right triangles.
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.…
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.
Rhombuses and hexagons
The diagonals of a rhombus intersect perpendicularly at their midpoint. This divides the rhombus into four equal right triangles, whose legs are the halves of the diagonals and whose hypotenuse is the side.
In a regular hexagon with side $\ell$, the radius equals the side, because it is made of $6$ equilateral triangles. The apothem $m$ is the altitude of one of them.
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…
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'…
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.
The Pythagorean theorem in space: boxes and pyramids
A box (rectangular prism) with dimensions $a$, $b$, $c$ has a space diagonal that goes from one vertex to the opposite vertex. It is calculated by using the Pythagorean theorem twice: first the diagonal of the base, and then the space diagonal.
In a regular pyramid with a square base there are three hidden right triangles:
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the height and half the side of the base, which give the slant height of the face;
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the slant height and half the side, which give the lateral edge;
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the height and half the diagonal of the base, which give the same thing.
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…
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.
Similar figures, ratio, and scale
Two figures are similar if they have the same shape: their angles are equal and their corresponding sides are proportional. The similarity ratio is $k = \dfrac{\text{measurement in the new figure}}{\text{measurement in the original figure}}$.
A scale of $1:500$ is a similarity ratio: $1$ cm on the plan is $500$ cm in real life.
Two similar antennas of $12$ m and $18$ m have $k = \dfrac{18}{12} = 1.5$, and all their linear measurements (anchoring distance included) are multiplied by $1.5$.
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…
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…
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?
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…
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.
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What is the total length of wire needed to secure the tower?
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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?
Perimeters and areas of similar figures
If $k$ is the similarity ratio, lengths (sides, perimeters, diagonals) are multiplied by $k$, but areas are multiplied by $k^2$.
You can also go the other way: if the area of one figure is half of the other, then $k^2 = \dfrac{1}{2}$ and $k = \sqrt{\dfrac{1}{2}} \approx 0.71$. If the perimeter triples, $k = 3$ and the area is multiplied by $9$ without calculating any side.
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 $k$…
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…
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.
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Calculate the actual dimensions of the desk in meters.
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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 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…
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.
Volumes of similar solids
In similar solids, volume stretches in three directions: length, width, and height. That is why it is multiplied by $k^3$.
For the pyramid, remember that $V = \dfrac{1}{3} \cdot \text{Base area} \cdot \text{height}$. With the height of the model ($k$ times the real one) you can calculate the volume directly, or use $k^3$ to check.
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…
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.
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…
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:
- a) The length of the slant edge of the actual pyramid (you will need to apply the Pythagorean theorem twice).
- 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).
Shadows and Thales' theorem
At the same time of day, the sun strikes all objects at the same angle. A pole and its shadow form a right triangle similar to the one formed by a building and its shadow. By Thales' theorem, the corresponding sides are proportional:
$$\frac{\text{height of the object}}{\text{shadow of the object}} = \frac{\text{height of the reference}}{\text{shadow of the reference}}$$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…
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…
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.
Triangles inside triangles
A line parallel to one side of a triangle creates a small triangle similar to the big one: a shelf parallel to the base of a bracket, a spotlight casting a shadow, the slope of a roof. You always look for the two similar figures and set up a proportion between their sides.
With projection from a light source it works the same way: the light and the object, and the light and the wall, form similar triangles. The important thing is to draw the correct distances. If the boy walks toward the pole, the height of the shadow on the wall changes.
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…
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:
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What is the length of this intermediate shelf?
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What is the length of the sloped wooden beam forming the hypotenuse of the full bookcase?
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…
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.
- a) Calculate the height of the flagpole knowing that the shadow of the boy's head coincides exactly with the top of the flagpole.
- b) If the boy walks $2\text{ m}$ towards the flagpole, how long will his shadow cast on the flagpole itself be?
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…
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.
- a) Calculate the actual length of the section of roof that covers a horizontal distance of $8\text{ m}$.
- 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).
The altitude theorem
In a right triangle, the altitude drawn from the right angle divides the hypotenuse into two segments, $m$ and $n$. The two small triangles are similar to each other and to the big one, and a simple relationship comes out of that.
This is useful when two rays or two cables form a right angle and the support point lies on the altitude: the ground is the hypotenuse and this theorem gives the altitude.
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…
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:
- a) The height $h$ at which the drone is flying using the altitude theorem.
- b) The distance from the drone to each of the two landing points (the legs of the triangle).
Back to the beginning: Wilkin's cell phone
The screen of Wilkin's cell phone is $6$ inches along the diagonal, that is, $6 \times 2.54 = 15.24$ cm. If the screen is $13.5$ cm tall, its width is a leg: $\sqrt{15.24^2 - 13.5^2} \approx 7.07$ cm. For Yaritza's antenna cable, the same type of calculation is done with the hypotenuse. In every case, you first look for the hidden right triangle.