Applications of the Pythagorean Theorem and Similarity
You are shopping for a TV and the box says "50-inch." Is that the width? The height? Neither. So how can you tell whether it will fit on your shelf? The trick in almost every real-life geometry problem is the same: find the right triangle hidden in the picture, then use the Pythagorean Theorem, similar triangles, or both.
The Diagonal of a Rectangle: Screens and Photos
In any rectangle, the width, the height and the diagonal form a right triangle. The diagonal is the longest side, so it is the hypotenuse.
If you know the two legs, add the squares to get $c^2$. If you know the hypotenuse and one leg, subtract to get the other leg: $b^2 = c^2 - a^2$.
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.
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.
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…
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.
The Diagonal of a Box
Can a long object fit inside a box if you tilt it from one corner to the opposite corner? That longest segment is the space diagonal. It hides two right triangles: first the diagonal of the floor, then a vertical triangle on top of it.
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…
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.
Wires, Ladders and Zip Lines
A tall pole, the ground, and a wire holding the pole up make a right triangle. The wire (or ladder, or zip line) is the hypotenuse because it connects the top to the ground and is the longest side.
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…
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?
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 arrival point is on another platform at a height of $3 \text{ m}$. What is the horizontal distance separating both structures?
Roofs and Isosceles Triangles
The end of a shed roof is an isosceles triangle. Draw its height and it lands at the midpoint of the base. Now you have two right triangles, each with a leg equal to half the base.
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 $5 \text{ meters}$…
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.
Rhombuses and Hexagons
The diagonals of a rhombus cross at right angles and cut each other in half. So a rhombus is made of four identical right triangles. Each leg is half of a diagonal, and the side of the rhombus is the hypotenuse.
A regular hexagon is made of six equilateral triangles that meet at the center. The apothem is the distance from the center to the midpoint of a side, so it is the height of one of those triangles.
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 intersect at their midpoint, forming four equal right-angled triangles).
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…
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.
Two Right Triangles Sharing a Side
Sometimes the picture holds two right triangles that share a leg. You can't solve either one alone, but writing the Pythagorean Theorem for both lets you compare them.
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…
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:
- a) The height of the cliff above sea level.
- b) The horizontal distance from the boat to the base of the cliff.
Shadows and Similar Triangles
At the same moment of the day, sunlight hits everything at the same angle. A person and a tree both make a right triangle with their shadows, and these two triangles have the same angles, so they are similar (AA). Matching sides of similar triangles are in the same ratio.
Some problems call this method Thales's theorem. In US classes it is simply AA similarity: two triangles with the same angles have proportional sides. The next section gives the version for a line drawn parallel to a side, the Triangle Proportionality Theorem.
The same idea works with a spotlight. A light on the ground, a person, and a wall make two similar triangles that share the angle at the light. The key is to measure both distances from the light.
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…
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 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…
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.
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…
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.
- a) Calculate the height of the flagpole knowing that the shadow of the teenager's head coincides exactly with the top of the flagpole.
- b) If the teenager walks $2\text{ m}$ towards the flagpole, what length will his shadow projected on the flagpole itself be?
Parallel Cuts and Slopes
Draw a line parallel to one side of a triangle. It cuts off a small triangle at the top that has the same angles as the whole triangle, so it is similar to it. This is how you find shelf lengths, roof panels and scaled-up copies.
A slope is a ratio that never changes, like "rises 5 for every 12 across." Any triangle with that ratio is similar to every other one.
If two triangles are similar, every pair of matching lengths (legs, hypotenuses, heights) has the same ratio $k$, called the scale factor. Example: a 6-8-10 triangle and a similar triangle with shortest side 9 have $k = \frac{9}{6} = 1.5$, so the other sides are $8 \cdot 1.5 = 12$ and $10 \cdot 1.5 = 15$.
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…
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:
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What is the length of this intermediate shelf?
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What is the length of the slanted batten that forms the hypotenuse of the complete shelving unit?
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…
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.
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What is the total length of wire needed to secure the tower?
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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?
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 $3\text{ m}$…
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.
- a) Calculate the real length of the roof section that covers $8\text{ m}$ of horizontal distance.
- 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).
The Altitude Theorem
Draw the altitude from the right angle of a right triangle down to the hypotenuse. It splits the hypotenuse into two pieces, $p$ and $q$, and creates two smaller right triangles. Both are similar to the original triangle (they share an acute angle with it).
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…
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:
- 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).
Scale Factor and Scale Drawings
Two figures are similar if one is an exact enlargement or reduction of the other. Every length is multiplied by the same number, the scale factor $k$:
$$k = \frac{\text{new length}}{\text{original length}}$$On a scale drawing (a map or floor plan), a scale such as $1:48$ means 1 unit on the drawing stands for 48 of the same unit in real life. So real length $=$ drawing length $\times$ 48. Convert at the end: $12$ in. $= 1$ ft.
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 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…
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?
Area and Volume of Similar Figures
If lengths are multiplied by $k$, perimeters are multiplied by $k$, but areas and volumes grow faster.
When you go from the real object to the model with scale $1:n$, you divide: lengths by $n$, areas by $n^2$, volumes by $n^3$.
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…
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.
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…
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.
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Calculate the real dimensions of the desk in feet.
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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 $\text{in}^2$?
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…
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.
Putting It All Together
Hard problems usually need two or three of the tools above. Make a plan: draw the figure, find the hidden right triangles, find the scale factor, and only then calculate.
When a problem gives an area ratio and asks for lengths, go backward: $k = \sqrt{\text{area ratio}}$.
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 $16\text{ ft}$.…
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:
- a) The length of the lateral edge of the real pyramid (you will need to apply Pythagoras' theorem twice).
- 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.…
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.
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 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:
- a) The perimeter of the rhombus.
- 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.