Applications of similarity
You already know what a similar figure is and how the ratio $k$ works. Now it is time to use it: with a plan you can work out the real size of a swimming pool without stepping on it, and with a shadow you can measure a tree without climbing it.
Scales: plans, maps and models
A plan, a map or a model is always similar to the real thing. The scale is the ratio of similarity, and it is written $1 : n$: each unit on the drawing represents $n$ real units.
The scale model of a new sports centre is built to a scale of $1:200$. If the Olympic swimming pool is $25 \text{ cm}$ long on the model, how many metres long will the…
The scale model of a new sports centre is built to a scale of $1:200$. If the Olympic swimming pool is $25 \text{ cm}$ long on the model, how many metres long will the pool be in reality?
On a floor plan of a house with a scale of $1:75$, the living room is rectangular with dimensions of $8$ cm long and $6$ cm wide. Calculate the actual surface area of…
On a floor plan of a house with a scale of $1:75$, the living room is rectangular with dimensions of $8$ cm long and $6$ cm wide. Calculate the actual surface area of the living room in square metres ($m^2$).
Screens, photos and similar objects
Two photos, two screens or any two rectangles are similar if they keep the same shape: the width and the height grow in the same proportion.
We have two similar rectangles. The first is $8 \text{ cm}$ wide and $12 \text{ cm}$ high. If the second has a width of $20 \text{ cm}$, what must its height be for it…
We have two similar rectangles. The first is $8 \text{ cm}$ wide and $12 \text{ cm}$ high. If the second has a width of $20 \text{ cm}$, what must its height be for it to maintain the same shape?
Two television screens are similar. The first has a diagonal of $32$ inches and its surface area is approximately $2800$ $cm^2$. If the second screen has a diagonal of…
Two television screens are similar. The first has a diagonal of $32$ inches and its surface area is approximately $2800$ $cm^2$. If the second screen has a diagonal of $48$ inches:
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What is the scale factor between the large screen and the small one?
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What surface area does the $48$ inch screen occupy?
Measuring heights with shadows
At the same time of day, the Sun makes the same angles with everything it lights. An object and its shadow, and another object and its shadow, form two similar right-angled triangles.
To measure the height of a tree in the school playground, we take advantage of a sunny day. A wooden batten placed vertically in the ground that is $1,5 \text{ m}$ high…
To measure the height of a tree in the school playground, we take advantage of a sunny day. A wooden batten placed vertically in the ground that is $1,5 \text{ m}$ high casts a shadow of $0,9 \text{ m}$. If at that same moment the shadow of the tree measures $6 \text{ m}$, what is the height of the tree?
A content creator is recording a video outdoors. To calculate the height of a modern sculpture, they place their tripod, which is $1,20$ metres high, so that it casts a…
A content creator is recording a video outdoors. To calculate the height of a modern sculpture, they place their tripod, which is $1,20$ metres high, so that it casts a shadow of $90$ cm on the ground. At that same moment, the shadow of the sculpture measures $4,5$ metres. How tall is the sculpture?
The altitude to the hypotenuse: the leg theorem and the altitude theorem
In a right-angled triangle, if you draw the altitude $h$ from the right angle to the hypotenuse, three triangles that are similar to each other appear: the large one and the two small ones into which it is divided.
In the right-angled triangle in the figure, the hypotenuse measures $25 \text{ cm}$. The height relative to the hypotenuse, $h$, divides it into two segments, $m$ and…
In the right-angled triangle in the figure, the hypotenuse measures $25 \text{ cm}$. The height relative to the hypotenuse, $h$, divides it into two segments, $m$ and $n$. If we know that leg $b$ measures $15 \text{ cm}$:
- a) Calculate the length of the projection $m$ using the Cathetus Theorem.
- b) Calculate the value of the height $h$ using the Altitude Theorem.
- c) Find the area and perimeter of the total triangle.