Similarity
When you zoom in on a photo on your phone, everything gets bigger in the same proportion: the photo doesn't get distorted. The two images are similar. Using this idea, about 2,600 years ago, Thales of Miletus measured the height of a pyramid using only its shadow and a stick. Today it is used in floor plans, maps, scale models, and 3D printers. How did Thales measure the pyramid? We'll solve it at the end.
Similar figures
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$k > 1$: enlargement. $\quad k < 1$: reduction.
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$k = 1$: the figures are congruent ($\cong$): same shape and same size. Congruence is the special case of similarity with $k = 1$.
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A scale is a ratio of similarity: a 1:50 plan has $k = \frac{1}{50}$.
Change the ratio: the sides are multiplied by $k$, but the area is multiplied by $k^2$.
A graphic designer is working on the logo for a new app. The original design is a rectangle measuring $8\text{ cm}$ wide and $12\text{ cm}$ high. For the loading screen,…
A graphic designer is working on the logo for a new app. The original design is a rectangle measuring $8\text{ cm}$ wide and $12\text{ cm}$ high. For the loading screen, he needs to create a larger version that is similar to the original. If the width of the enlarged version must be $20\text{ cm}$, what will be the height of the logo on that screen?
On the floor plan of a house, the scale used is $1:50$. If the living room has a length of $12\text{ cm}$ in the drawing, calculate the actual length of this room in…
On the floor plan of a house, the scale used is $1:50$. If the living room has a length of $12\text{ cm}$ in the drawing, calculate the actual length of this room in meters.
A graphic designer is working on the logo for a new clothing brand. The original design is a quadrilateral with side lengths $6\text{ cm}$, $8\text{ cm}$, $9\text{ cm}$…
A graphic designer is working on the logo for a new clothing brand. The original design is a quadrilateral with side lengths $6\text{ cm}$, $8\text{ cm}$, $9\text{ cm}$ and $12\text{ cm}$. The client asks for a larger version for a billboard, where the side that previously measured $9\text{ cm}$ must now measure $45\text{ cm}$.
- a) Calculate the scale factor $k$ between the billboard logo and the original.
- b) Determine the lengths of the other three sides in the billboard logo.
- c) If the perimetre of the original logo is $35\text{ cm}$, what will be the perimetre of the billboard logo without adding its sides one by one?
Similarity and congruence postulates
To know whether two triangles are similar you don't need to check everything. One of these postulates is enough:
| Postulate | What you need to check |
|---|---|
| AA | Two equal angles (the third one is also equal, since they add up to 180°). |
| SSS | All three sides in the same ratio: $\frac{a'}{a} = \frac{b'}{b} = \frac{c'}{c}$. |
| SAS | One equal angle and the two sides that form it proportional: $\angle A = \angle A'$ and $\frac{b'}{b} = \frac{c'}{c}$. |
Each one works because it fixes the shape of the triangle: with two angles, or three ratios, or an angle and its sides, only one triangle of that shape can be drawn; the size is the only thing that can vary.
If, in addition, the ratio is $k = 1$, the triangles are congruent. Then the congruence postulates are used:
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SSS: three equal sides.
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SAS: two sides and the included angle equal.
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ASA: two angles and the included side equal.
State whether the following pairs of triangles are similar and justify which criterion you used (AAA, SSS or SAS): Triangle 1: Angles of $40^\circ$ and $70^\circ$.…
State whether the following pairs of triangles are similar and justify which criterion you used (AAA, SSS or SAS):
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Triangle 1: Angles of $40^\circ$ and $70^\circ$.
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Triangle 2: Angles of $70^\circ$ and $70^\circ$.
Consider an isosceles triangle $ABC$ where the unequal side $BC$ measures $12\text{ cm}$ and the equal sides measure $10\text{ cm}$ each. Inscribed in it is a square…
Consider an isosceles triangle $ABC$ where the unequal side $BC$ measures $12\text{ cm}$ and the equal sides measure $10\text{ cm}$ each. Inscribed in it is a square whose base lies on side $BC$ of the triangle and its other two vertices touch sides $AB$ and $AC$.
Use the similarity between the original triangle and the small triangle that lies above the square to calculate the side length of the square.
Thales' theorem
Proof
Step 1: the triangle in Thales position. Let $ABC$ be a triangle and $DE$ a line parallel to $BC$, with $D$ on $AB$ and $E$ on $AC$.
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Triangles $ADE$ and $ABC$ share angle $A$.
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$\angle ADE = \angle ABC$, because they are corresponding angles between the parallels $DE$ and $BC$ cut by the transversal $AB$.
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By the AA postulate, $\triangle ADE \sim \triangle ABC$, and their corresponding sides are proportional:
Step 2: several parallels. In the figure, draw through $A$ a line parallel to $A'C'$. It meets the parallels through $B$ and through $C$ at $P$ and $Q$. The quadrilaterals $AA'B'P$ and $PB'C'Q$ have parallel opposite sides, so $AP = A'B'$ and $PQ = B'C'$. In triangle $ACQ$, the line $BP$ is parallel to $CQ$, and by step 1:
$$\frac{AB}{AC} = \frac{AP}{AQ}$$Subtracting parts ($BC = AC - AB$ and $PQ = AQ - AP$) we get $\dfrac{AB}{BC} = \dfrac{AP}{PQ} = \dfrac{A'B'}{B'C'}$. ∎
A very useful case is triangles in Thales position: a small triangle inside a large one, sharing an angle and with the opposite side parallel. They are similar (step 1).
Two triangles are positioned in Thales's configuration. Knowing that the horizontal segments are parallel and measure $6\text{ cm}$ and $10\text{ cm}$, and that the…
Two triangles are positioned in Thales's configuration. Knowing that the horizontal segments are parallel and measure $6\text{ cm}$ and $10\text{ cm}$, and that the vertical segment of the smaller triangle measures $9\text{ cm}$, calculate the value of the unknown vertical segment denoted by $x$ of the larger triangle.
In the following figure, lines $r$, $s$ and $t$ are parallel. Use Thales' theorem to calculate the unknown lengths $x$ and $y$, expressed in centimeters.
In the following figure, lines $r$, $s$ and $t$ are parallel. Use Thales' theorem to calculate the unknown lengths $x$ and $y$, expressed in centimeters.
Two cities, A and B, are separated by a river with parallel banks. To calculate the distance between a point $P$ on the bank of city A and a point $Q$ on the bank of…
Two cities, A and B, are separated by a river with parallel banks. To calculate the distance between a point $P$ on the bank of city A and a point $Q$ on the bank of city B (located directly opposite), a surveyor travels $60\text{ m}$ along the bank of city A to a point $R$. From $R$, he walks another $20\text{ m}$ in the same direction to a point $S$. At $S$, he turns $90^\circ$ and moves away from the river until points $Q$ and $R$ are aligned with his current position ($T$). If the distance $ST$ is $15\text{ m}$:
- a) Calculate the width of the river ($PQ$).
- b) Calculate the straight-line distance between the surveyor at point $T$ and point $Q$ on the other side of the river.
The altitude to the hypotenuse
In a right triangle, the altitude drawn from the right angle to the hypotenuse divides it into two smaller right triangles. All three triangles are similar.
We call the legs $a$ and $b$, the hypotenuse $c$, the altitude $h$, and the segments of the hypotenuse $m$ and $n$ ($BD = m$, $AD = n$).
Why they are similar. All three triangles have a right angle.
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$\triangle ABC$ and $\triangle CBD$ share angle $B$.
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$\triangle ABC$ and $\triangle ACD$ share angle $A$.
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By AA, $\triangle ABC \sim \triangle CBD \sim \triangle ACD$. The two small ones are also similar to each other, and in them $\angle ACD = \angle B$.
Observe the right-angled triangle in the figure. The altitude $h$ to the hypotenuse divides the original triangle into two smaller triangles. a) Justify why the three…
Observe the right-angled triangle in the figure. The altitude $h$ to the hypotenuse divides the original triangle into two smaller triangles.
- a) Justify why the three triangles (the original and the two small ones) are similar to each other.
- b) If the segments into which the hypotenuse is divided measure $m = 4\text{ cm}$ and $n = 9\text{ cm}$, calculate the altitude $h$ using the similarity of the two small triangles.
- c) Calculate the area of the original triangle.
An architect is designing the facade of a modern auditorium. He has drawn a main right-angled triangle and, to improve acoustics, needs to draw the altitude to the…
An architect is designing the facade of a modern auditorium. He has drawn a main right-angled triangle and, to improve acoustics, needs to draw the altitude to the hypotenuse. This altitude divides the original triangle into two smaller triangles. Knowing that the legs of the original triangle measure $18\text{ m}$ and $24\text{ m}$, use triangle similarity to calculate:
- a) The length of the altitude drawn to the hypotenuse.
- b) The lengths of the two segments into which the hypotenuse is divided.
Measuring heights with shadows
At the same time of day, the Sun's rays arrive parallel. That is why an object and its shadow form a right triangle that is similar (AA postulate) to the one formed by any other object and its shadow.
The same reasoning works with a mirror on the ground: the ray is reflected at equal angles, and again two similar right triangles are formed.
A 3rd ESO student is $1.60\text{ m}$ tall. At a certain time of day, they cast a shadow on the ground of $2\text{ m}$. If right next to them, a lamppost casts a shadow…
A 3rd ESO student is $1.60\text{ m}$ tall. At a certain time of day, they cast a shadow on the ground of $2\text{ m}$. If right next to them, a lamppost casts a shadow of $5\text{ m}$, what is the height of the lamppost?
To calculate the height of a communication tower that casts a very long shadow, Marcos, who is $1.70\text{ m}$ tall, stands so that the tip of his shadow coincides with…
To calculate the height of a communication tower that casts a very long shadow, Marcos, who is $1.70\text{ m}$ tall, stands so that the tip of his shadow coincides with the tip of the tower's shadow. If Marcos is $25\text{ m}$ from the base of the tower and his shadow measures $2.5\text{ m}$, answer:
- a) Why can we state that the triangles formed by Marcos and the tower with their respective shadows are similar? Mention the similarity criterion used.
- b) Calculate the height of the tower.
To measure the height of a cooling tower at a power station, a team of engineers uses a mirror placed on the ground. The engineer stands such that, looking into the…
To measure the height of a cooling tower at a power station, a team of engineers uses a mirror placed on the ground. The engineer stands such that, looking into the mirror, he sees the top of the tower reflected. The mirror is $45\text{ m}$ from the base of the tower. The engineer, whose eyes are at a height of $1.75\text{ m}$, is $1.5\text{ m}$ from the mirror. Between the mirror and the tower there is a safety fence of height $5\text{ m}$.
What is the minimum distance from the mirror that the fence must be positioned so that it does not block the reflection of the top of the tower from the engineer's eyes?
Perimeters, areas, and volumes
If the lengths are multiplied by $k$:
A pizzeria offers two sizes of round pizza that are similar. The "Small" pizza has a radius of $10\text{ cm}$ and an area of approximately $314\text{ cm}^2$. If the…
A pizzeria offers two sizes of round pizza that are similar. The "Small" pizza has a radius of $10\text{ cm}$ and an area of approximately $314\text{ cm}^2$. If the "Family" pizza has a radius of $20\text{ cm}$, what will its area be? (Remember the relationship between the scale factor $k$ and the areas).
Two rectangles are similar. The first has a base of $12\text{ cm}$ and an area of $96\text{ cm}^2$. The second rectangle has an area of $150\text{ cm}^2$. a) Calculate…
Two rectangles are similar. The first has a base of $12\text{ cm}$ and an area of $96\text{ cm}^2$. The second rectangle has an area of $150\text{ cm}^2$.
- a) Calculate the scale factor between the second and first rectangle. (Hint: use the relationship between areas).
- b) Calculate the dimensions (base and height) of the second rectangle.
- c) What is the ratio between their perimetres?
A 3D printing company has created a scale model of a circular square. In the model, the area of the square is $450\text{ cm}^2$. The client requests a second model where…
A 3D printing company has created a scale model of a circular square. In the model, the area of the square is $450\text{ cm}^2$. The client requests a second model where the perimetre is $20\%$ larger than that of the first model.
- a) What will be the area of the square in this second model?
- b) If the actual square has a radius of $30\text{ m}$, what scale was used in the first model? (Express the scale in the format $1:n$).
Back to the beginning: Thales stuck a vertical stick in the ground and measured its shadow at the same time as the pyramid's shadow. Since the Sun's rays are parallel, the triangles are similar, and he set up $\frac{\text{pyramid height}}{\text{pyramid shadow}} = \frac{\text{stick height}}{\text{stick shadow}}$.