Similarity
How tall is your school's flagpole? You can't climb it, and you don't have a 40-foot tape measure. But you do have a yardstick and the sun. When you zoom in on a photo, everything grows by the same factor and nothing gets distorted. The two images are similar. The same idea lets you measure a flagpole from its shadow, read a map, build a scale model or print in 3D.
Similar figures and the scale factor
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$k > 1$: an enlargement. $\quad 0 < k < 1$: a reduction. $\quad k = 1$: congruent figures.
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A dilation with scale factor $k$ produces a similar figure.
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Always write the ratio as new ÷ original, so you know which way you are going.
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Perimeters scale the same way as sides. If every side is multiplied by $k$, the perimeter is multiplied by $k$ too.
Drag the slider. 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 the height of the logo be on that screen?
A graphic designer is working on the logo for a new clothing brand. The original design is a quadrilateral with sides measuring $6\text{ cm}$, $8\text{ cm}$,…
A graphic designer is working on the logo for a new clothing brand. The original design is a quadrilateral with sides measuring $6\text{ cm}$, $8\text{ cm}$, $9\text{ cm}$ and $12\text{ cm}$. The client asks for a larger version for an advertising poster, where the side that previously measured $9\text{ cm}$ must now measure $45\text{ cm}$.
- a) Calculate the scale factor $k$ between the poster logo and the original.
- b) Determine the length of the other three sides in the poster logo.
- c) If the perimeter of the original logo is $35\text{ cm}$, what will the perimeter of the poster logo be without adding its sides one by one?
Scale drawings and maps
A scale like $1:50$ is a scale factor: $k = \frac{1}{50}$. One unit on the drawing stands for $50$ of the same units in real life.
To use a scale: (1) convert the lengths to the same unit, (2) multiply by $50$ to go from drawing to real life, or divide by $50$ to go the other way.
Areas follow $k^2$, not $k$. On a $1:75$ plan, a real area is $75^2 = 5{,}625$ times the area on the plan (see the last section of this page).
On a house plan, the scale used is $1:50$. If the living room has a length of $12\text{ in}$ on the drawing, calculate how many feet the actual length of this room…
On a house plan, the scale used is $1:50$. If the living room has a length of $12\text{ in}$ on the drawing, calculate how many feet the actual length of this room measures.
A road map has a scale of $1:200.000$. If the actual distance between two towns is $36\text{ mi}$, at what distance will they be on the map expressed in inches?
A road map has a scale of $1:200.000$. If the actual distance between two towns is $36\text{ mi}$, at what distance will they be on the map expressed in inches?
On a house plan with a scale of $1:75$, the living room has a rectangular shape with dimensions of $8\text{ in}$ long and $6\text{ in}$ wide. a) Calculate the actual…
On a house plan with a scale of $1:75$, the living room has a rectangular shape with dimensions of $8\text{ in}$ long and $6\text{ in}$ wide.
- a) Calculate the actual dimensions of the living room in feet.
- b) Find the area of the living room on the plan (in $cm^2$) and the actual area of the living room (in $m^2$).
- c) Check that the ratio between the actual area and the area on the plan is equal to the square of the scale factor ($k^2$).
Similar triangles: AA, SSS and SAS
To show two triangles are similar you do not need to check everything. One of these tests is enough.
| Test | What to check |
|---|---|
| AA (angle-angle) | Two angles of one triangle are congruent to two angles of the other. The third angles match too, since angles add up to $180^\circ$. |
| SSS (side-side-side) | All three pairs of sides are proportional: $\frac{a'}{a} = \frac{b'}{b} = \frac{c'}{c}$. |
| SAS (side-angle-side) | One pair of angles is congruent and the two sides forming that angle are proportional: $\angle A \cong \angle A'$ and $\frac{b'}{b} = \frac{c'}{c}$. |
Some problems list a test called AAA. It is the same as AA: once two angles match, the third has to match as well.
State whether the following pairs of triangles are similar and justify which criterion you have 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 have 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 $70^\circ$.
The Triangle Proportionality Theorem
The same idea works for three or more parallel lines cut by two transversals: the pieces on one transversal are proportional to the pieces on the other. Some problems call this Thales's theorem.
Another very useful picture is the hourglass: two lines cross at a point, and two parallel segments close off the "bow tie." The two triangles are similar (vertical angles plus alternate interior angles).
Two triangles are placed in Thales's position, as shown in the following figure. Knowing that the horizontal segments are parallel, calculate the value of the unknown…
Two triangles are placed in Thales's position, as shown in the following figure. Knowing that the horizontal segments are parallel, calculate the value of the unknown segment denoted by $x$.
In the following figure, lines $r$, $s$ and $t$ are parallel. Use Thales's theorem to calculate the unknown lengths $x$ and $y$ expressed in inches.
In the following figure, lines $r$, $s$ and $t$ are parallel. Use Thales's theorem to calculate the unknown lengths $x$ and $y$ expressed in inches.
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 walks $60\text{ m}$ 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 walks 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.
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 rests 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 above the square to calculate the side of the square.
Indirect measurement: shadows and mirrors
At the same time of day, the sun's rays are parallel. So an object and its shadow form a right triangle similar to the one made by any other object and its shadow: both have a right angle and the same sun angle (AA).
Mirror trick. Put a mirror flat on the ground between you and a tall object. When you can see the top of the object in the mirror, light is reflecting with the angle of incidence equal to the angle of reflection. So your right triangle and the object's right triangle share an acute angle, and they are similar by AA.
A 10th grade student is $1,60\text{ m}$ tall. At a certain time of day, he casts a shadow on the ground of $2\text{ m}$. If right next to him, a streetlight casts a…
A 10th grade student is $1,60\text{ m}$ tall. At a certain time of day, he casts a shadow on the ground of $2\text{ m}$. If right next to him, a streetlight casts a shadow of $5\text{ m}$, what is the height of the streetlight?
To calculate the height of a communications tower that casts a very long shadow, Ethan, who is $1,70\text{ m}$ tall, stands so that the end of his own shadow coincides…
To calculate the height of a communications tower that casts a very long shadow, Ethan, who is $1,70\text{ m}$ tall, stands so that the end of his own shadow coincides with the end of the tower's shadow. If Ethan 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 Ethan 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 so 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 so that, looking into the mirror, he sees the reflection of the top of the tower. 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 security fence $5\text{ m}$ high.
What is the minimum distance the fence must be from the mirror so that it does not hide the reflection of the top of the tower from the engineer's eyes?
The altitude to the hypotenuse
Draw the altitude from the right angle of a right triangle to the hypotenuse. It cuts the triangle into two smaller right triangles. All three triangles are similar.
Why? $\triangle ACB$, $\triangle ADC$ and $\triangle CDB$ are all right triangles. $\triangle ADC$ shares $\angle A$ with the big triangle, and $\triangle CDB$ shares $\angle B$. By AA, all three are similar.
Look at the right triangle in the figure. The altitude $h$ to the hypotenuse divides the original triangle into two smaller triangles. a) Justify why the three triangles…
Look at the right 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 smaller 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 smaller 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 triangle and, to improve the acoustics, he needs to draw the altitude to the…
An architect is designing the facade of a modern auditorium. He has drawn a main right triangle and, to improve the acoustics, he needs to draw the altitude to the hypotenuse. This altitude divides the original triangle into two smaller triangles. Given that the legs of the original triangle measure $18\text{ m}$ and $24\text{ m}$, use the similarity of triangles 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.
Projectors, light beams and cones
A light that shines from one point (a projector, a flashlight, a drone) makes a triangle (from the side) or a cone. Any cross-section parallel to the screen is similar to the image on the screen. The scale factor is the ratio of the distances measured from the light source.
Here $d$ is the distance from the source to the object and $D$ is the distance from the source to the screen. Then
$$\frac{w}{W} = \frac{d}{D} \qquad \text{and the areas satisfy} \qquad \frac{\text{area on screen}}{\text{object area}} = \left(\frac{D}{d}\right)^2.$$A movie projector is located $6\text{ meters}$ from a screen. The light beam projects an image that has a height of $2,4\text{ meters}$ on the screen. a) If we place an…
A movie projector is located $6\text{ meters}$ from a screen. The light beam projects an image that has a height of $2,4\text{ meters}$ on the screen.
- a) If we place an obstacle $40\text{ cm}$ high parallel to the screen to create a shadow puppet, at what distance from the projector must we place it so that its shadow occupies exactly the full height of the screen?
- b) If we move the projector away to $9\text{ meters}$ from the screen, what will the new height of the projected image be?
A slide is projected onto a screen located at a distance of $6\text{ m}$. The size of the projected image is $1,8\text{ m}$ high by $2,4\text{ m}$ wide. a) If the…
A slide is projected onto a screen located at a distance of $6\text{ m}$. The size of the projected image is $1,8\text{ m}$ high by $2,4\text{ m}$ wide.
- a) If the original slide has dimensions of $24\text{ mm}$ by $32\text{ mm}$, at what distance from the slide is the projector's light source?
- b) If we move the screen so that it is $9\text{ m}$ from the projector, how many times will the area of the image increase compared to the previous projection?
A drone projects a conical beam of light onto flat ground. At a height of $10\text{ m}$, the beam illuminates a circle with a radius of $4\text{ m}$. The drone descends…
A drone projects a conical beam of light onto flat ground. At a height of $10\text{ m}$, the beam illuminates a circle with a radius of $4\text{ m}$. The drone descends to a height where the illuminated area is reduced to exactly a quarter of the original.
Draw the diagram in your notebook.
- a) At what height is the drone now?
- b) If there is an object $1,5\text{ m}$ high in the center of the illuminated area, what will be the radius of the shadow cast by this object when the drone was at a height of $10\text{ m}$?
Perimeter, area and volume
If the lengths of a figure are multiplied by $k$:
To go the other way, from areas to a scale factor, take a square root: $k = \sqrt{\dfrac{\text{area}'}{\text{area}}}$. From volumes, take a cube root.
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 the first rectangle. (Hint: use the relationship between areas).
- b) Calculate the dimensions (base and height) of the second rectangle.
- c) What is the ratio of their perimeters?
A furniture company manufactures regular hexagonal coffee tables. They have two models: the "Mini" model has a side of $30\text{ cm}$ and its glass top costs…
A furniture company manufactures regular hexagonal coffee tables. They have two models: the "Mini" model has a side of $30\text{ cm}$ and its glass top costs $45\text{ \$}$. The "Grand Lounge" model is similar to the previous one and has a side of $60\text{ cm}$.
- a) What is the scale factor between the large and the small model?
- b) Knowing that the price of the glass is proportional to its surface area, how much will the glass top of the "Grand Lounge" model cost?
- c) If $180\text{ cm}$ of metal tape is needed to edge the small table, how much tape will be needed for the large table?
A 3D printing company has created a scale model of a circular plaza. In the model, the area of the plaza is $450\text{ cm}^2$. The client requests a second model where…
A 3D printing company has created a scale model of a circular plaza. In the model, the area of the plaza is $450\text{ cm}^2$. The client requests a second model where the perimeter is $20\%$ greater than that of the first model.
- a) What will the area of the plaza be in this second model?
- b) If the real plaza has a radius of $30\text{ m}$, what is the scale used in the first model? (Express the scale in the format $1:n$).
On a map of a forest area, a fire has affected a trapezoidal area. The bases of the trapezoid on the map measure $4\text{ cm}$ and $7\text{ cm}$, and its height is…
On a map of a forest area, a fire has affected a trapezoidal area. The bases of the trapezoid on the map measure $4\text{ cm}$ and $7\text{ cm}$, and its height is $3\text{ cm}$. It is known that the actual affected area is $16,5\text{ hectares}$ ($1\text{ ha} = 10\,000\text{ m}^2$).
- a) Calculate the scale of the map.
- b) If it is decided to reforest an adjacent area which on the map is a triangle similar to the one that would be formed by extending the non-parallel sides of the trapezoid, and whose base is the shorter base of the trapezoid, what will be the actual area of this reforestation zone?
Back to the flagpole: push a yardstick vertically into the ground and measure its shadow right when you measure the flagpole's shadow. The sun's rays are parallel, so the two right triangles are similar by AA. Then $\dfrac{\text{flagpole height}}{\text{flagpole shadow}} = \dfrac{\text{yardstick height}}{\text{yardstick shadow}}$, and you have the height without climbing anything.