Similar Triangles
Mateo is standing at the foot of the school flagpole with a tape measure that is far too short to reach the top. He notices that his own shadow and the flagpole's shadow are both stretched across the blacktop. Could a few measurements on the ground, plus a little math, tell him how tall the pole is?
Similar triangles and the scale factor
Think about zooming in on a photo. The picture gets bigger, but its shape does not change. Triangles can do the same thing.
The ratio of any pair of corresponding sides is the same number. We call it the scale factor $k$:
$$k = \dfrac{\text{side of the new triangle}}{\text{corresponding side of the original}}$$If $k > 1$ the new triangle is larger. If $0 < k < 1$ it is smaller. If $k = 1$ the triangles are congruent.
Often you do not get $k$ handed to you. You find it from one pair of corresponding sides, then use it for the rest.
Here is why "same steepness" is a similar-triangle idea. A wheelchair ramp, a roof, or a staircase makes a right triangle with a "rise" and a "run." If you keep the same angle but make the ramp longer, you get a bigger similar triangle. The rise and run both grow by the same scale factor.
Calculate the measure of the sides of a triangle similar to another whose sides measure $5$ in, $7$ in and $9$ in, if the scale factor is $k = 3$. What will the new…
Calculate the measure of the sides of a triangle similar to another whose sides measure $5$ in, $7$ in and $9$ in, if the scale factor is $k = 3$. What will the new triangle be like, larger or smaller than the original?
Calculate the scale factor $k$ that converts the small triangle into the large triangle, given that they are similar. Once $k$ is found, indicate how much the side…
Calculate the scale factor $k$ that converts the small triangle into the large triangle, given that they are similar. Once $k$ is found, indicate how much the side marked with the letter $x$ must measure.
The town council wants to install a wheelchair access ramp. The original design is a right triangle with a base of $4$ feet and a height of $0,8$ feet. By regulation,…
The town council wants to install a wheelchair access ramp. The original design is a right triangle with a base of $4$ feet and a height of $0,8$ feet. By regulation, they must extend the base to $6$ feet while maintaining the same inclination (angle).
a) What height will the new ramp have?
b) If $4,08$ feet of rolling surface (hypotenuse) were needed for the original ramp, how many feet will be needed for the extended ramp?
The AA shortcut
Checking all three angles and all three sides of two triangles would take forever. Luckily, you only need to check a few things. The first shortcut uses angles.
Remember that the three angles of any triangle add up to $180^\circ$. If two angles match, the third one has no choice: it must match too.
Why does it work? Two angles fix the third, so the "shape" of the triangle is locked in. The only thing left to change is the size.
Two special cases come up a lot:
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Right triangles. Every right triangle already has one $90^\circ$ angle. So two right triangles with one equal acute angle (say, both have a $37^\circ$ angle) are similar by AA.
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Isosceles triangles. If two isosceles triangles have the same angle between their equal sides, their base angles are equal too, because each base angle is $\dfrac{180^\circ - \text{top angle}}{2}$. So they are similar.
Two triangles are similar if their angles are equal. Check if the following two triangles are similar by calculating the missing angle in each one: Triangle A has two…
Two triangles are similar if their angles are equal. Check if the following two triangles are similar by calculating the missing angle in each one:
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Triangle A has two angles measuring $45^\circ$ and $75^\circ$.
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Triangle B has two angles measuring $75^\circ$ and $60^\circ$.
Remember that the sum of the interior angles of any triangle is always $180^\circ$.
Two right triangles have an equal acute angle of $37^\circ$. Can we state that they are similar? Justify your answer using the similarity criteria for triangles.
Two right triangles have an equal acute angle of $37^\circ$. Can we state that they are similar? Justify your answer using the similarity criteria for triangles.
Two 7th grade students, Ashley and Marcus, are checking if two triangular sails from their dinghies are similar. Ashley's sail has two angles measuring $65^\circ$ and…
Two 7th grade students, Ashley and Marcus, are checking if two triangular sails from their dinghies are similar. Ashley's sail has two angles measuring $65^\circ$ and $45^\circ$. Marcus's sail has one angle of $45^\circ$ and another of $70^\circ$.
a) Calculate the third angle of each of the sails.
b) Reason, using the similarity criteria, whether the two sails have the same shape (are similar).
SSS and SAS
Sometimes you know lengths instead of angles. There are two more shortcuts.
Why? With SSS, one common ratio means that the whole triangle was just scaled. With SAS, the angle fixes the shape at that corner, and proportional sides fix how far it opens.
Check if these two triangles are similar using the proportional sides criterion (SSS criterion). To do this, divide the corresponding sides (the largest by the largest,…
Check if these two triangles are similar using the proportional sides criterion (SSS criterion). To do this, divide the corresponding sides (the largest by the largest, the middle by the middle and the smallest by the smallest) and see if you get the same result.
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Triangle 1: Sides of $6$ in, $9$ in and $12$ in.
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Triangle 2: Sides of $4$ in, $6$ in and $8$ in.
We have two isosceles triangles. The first has an unequal angle of $40^\circ$ and its equal sides measure $10\text{ cm}$ each. The second also has an unequal angle of…
We have two isosceles triangles. The first has an unequal angle of $40^\circ$ and its equal sides measure $10\text{ cm}$ each. The second also has an unequal angle of $40^\circ$ but its equal sides measure $15\text{ cm}$.
a) Are these two triangles similar? Justify your answer by citing the similarity criterion applied.
b) What is the scale factor between the second and the first?
c) If the base of the first triangle measures $6,84\text{ cm}$, what will the base of the second measure?
A parallel line inside a triangle
Shelves in a triangular bookcase, or beams in a roof truss, are often parallel to the base. Each parallel piece cuts off a smaller triangle that looks just like the whole.
Why? Parallel lines cut the sides at equal (corresponding) angles, so $\angle P = \angle B$ and $\angle Q = \angle C$. The angle at $A$ is shared. That is AA, so the triangles are similar and their sides are proportional.
Some problems describe this picture as being "in Thales's position." That is just another name for the same configuration.
Emma has drawn two triangles in Thales's position. Given that the horizontal segments are parallel, calculate the measure of side $x$ using the proportion between their…
Emma has drawn two triangles in Thales's position. Given that the horizontal segments are parallel, calculate the measure of side $x$ using the proportion between their corresponding sides.
Use Thales's theorem to find the value of $x$ in the following figure, where the shelves of the triangular shelving unit are parallel to the base.
Use Thales's theorem to find the value of $x$ in the following figure, where the shelves of the triangular shelving unit are parallel to the base.
In the design of a metal structure for a roof, two parallel beams are used to reinforce the main truss. Given that the triangles formed are in Thales's position,…
In the design of a metal structure for a roof, two parallel beams are used to reinforce the main truss. Given that the triangles formed are in Thales's position, calculate the length of the segments marked as $x$ and $y$ in the following diagram:
(Note: At the base, the small segment measures $4\text{ m}$ and the total $10\text{ m}$. The parallel reinforcement measures $5\text{ m}$. Find $x$, the remaining base, and $y$, the right side of the reinforcement).
Part and whole: finding the missing piece
Many problems give you a piece of a side but ask about the whole side, or the other way around. Two habits help:
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Always decide which segment is the whole side of the big triangle.
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If a problem asks about the leftover piece, subtract at the end.
Calculate the distance between points $B$ and $E$ in the following figure, given that segments $BC$ and $DE$ are parallel. The known data are: $AB = 6\text{ cm}$,…
Calculate the distance between points $B$ and $E$ in the following figure, given that segments $BC$ and $DE$ are parallel. The known data are: $AB = 6\text{ cm}$, $BC = 5\text{ cm}$ and $DE = 12,5\text{ cm}$.
In a triangle $ABC$, side $AB$ measures $12$ in. and side $AC$ measures $15$ in. We draw a segment $PQ$ parallel to side $BC$, such that point $P$ is on $AB$ and point…
In a triangle $ABC$, side $AB$ measures $12$ in. and side $AC$ measures $15$ in. We draw a segment $PQ$ parallel to side $BC$, such that point $P$ is on $AB$ and point $Q$ is on $AC$. If segment $AP$ measures a third of what segment $PB$ measures, calculate the lengths of segments $AP$ and $AQ$. Justify your answer based on Thales's theorem.
Shadows and indirect measurement
You can measure something you cannot reach by building a similar triangle you can reach.
At the same moment of the day, the sun's rays are parallel. So every vertical object (a person, a pole, a tree) makes the same angle with the ground, and each one forms a right angle with the ground. Two equal angles means AA, so the triangles formed by each object, its shadow, and a sun ray are similar.
Back to the start: Mateo's shadow stretches $4$ ft for every $5$ ft of height. That is $1.25$ ft of height per foot of shadow. The pole's shadow is $24$ ft, so the pole is $24 \times 1.25 = 30$ ft tall, with nothing but a short tape measure.
A second common setup: the tip of the shadow of a short stick lands exactly at the tip of a tall object's shadow. Then both triangles share the same corner at the shadow tip, so they share an angle. They both have a right angle, so they are similar by AA. Be careful here, because the stick sits inside the big triangle.
A tree casts a shadow of $12$ feet. At that same moment, a walking stick of $1,5$ feet driven vertically into the ground casts a shadow of $2$ feet. What is the height…
A tree casts a shadow of $12$ feet. At that same moment, a walking stick of $1,5$ feet driven vertically into the ground casts a shadow of $2$ feet. What is the height of the tree? Explain why the triangles formed by the tree and the stick with their shadows are similar.
To measure the height of a basketball hoop, a group of students uses a pole of $1,5$ feet placed vertically. At the same instant that the shadow of the pole measures…
To measure the height of a basketball hoop, a group of students uses a pole of $1,5$ feet placed vertically. At the same instant that the shadow of the pole measures $1,2$ feet, the shadow of the hoop projected on the ground measures $2,4$ feet. Calculate the height of the hoop taking advantage of the similarity of the triangles formed by the sun.
A group of 7th grade students wants to measure the height of the communication tower in their city. To do this, they use a vertical pole $2,5$ feet high. They place the…
A group of 7th grade students wants to measure the height of the communication tower in their city. To do this, they use a vertical pole $2,5$ feet high. They place the pole in such a way that, at a given moment, the end of the tower's shadow and that of the pole coincide at the same point on the ground. If the distance between the base of the pole and the point where the shadows end is $3,2$ feet, and the distance between the base of the tower and the base of the pole is $18,4$ feet, calculate the height of the tower.
Bow ties and lines of sight
Surveyors measure rivers, ponds, and canyons without crossing them. Two typical setups use similar triangles.
The bow tie. Two lines cross at a point $C$. One triangle sits on each side of $C$, and the "outer" sides $AB$ and $ED$ are parallel.
Why are the triangles similar?
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The angles at $C$ are vertical angles, so they are equal.
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$AB \parallel ED$, so the angles $\angle A$ and $\angle E$ are equal. They are alternate interior angles formed by the transversal $AE$. The same is true of $\angle B$ and $\angle D$.
That gives AA: $\triangle ABC \sim \triangle EDC$.
Lines of sight. When you look at a point past an edge, your eye, the edge, and the point are on one straight line. Together with the vertical wall and the standing person, this makes two right triangles that share the angle at your eye.
To calculate the width of a river without crossing it, a surveyor places several stakes as shown in the figure. The triangles formed, $ABC$ and $CDE$, are similar…
To calculate the width of a river without crossing it, a surveyor places several stakes as shown in the figure. The triangles formed, $ABC$ and $CDE$, are similar because the banks are parallel and the lines of sight cross at point $C$. If the measurements taken are $AB = 24$ ft, $BC = 30$ ft, and $CD = 15$ ft, determine the width of the river ($ED$) and explain which similarity criterion you have applied.
Daniel wants to measure the depth of a circular well of $1,6$ feet in diameter. To do this, he moves $0,8$ feet away from the edge until, from a height of $1,75$ feet…
Daniel wants to measure the depth of a circular well of $1,6$ feet in diameter. To do this, he moves $0,8$ feet away from the edge until, from a height of $1,75$ feet (his eye level), he manages to see exactly the edge of the bottom of the well. Use the similarity of triangles to find the depth of the well.
Perimeter and area of similar triangles
When you scale a figure by $k$, lengths multiply by $k$. But area does not multiply by $k$.
Picture a rectangle that is $2$ by $3$. Its area is $6$. Double both sides ($k = 2$): the new rectangle is $4$ by $6$, with area $24$. You can fit $4$ copies of the original inside it.
Why $k^2$? Area is length times width, and both lengths are multiplied by $k$. So you multiply by $k \times k = k^2$.
Light spreading from a single point (a projector, a flashlight, a drone camera) also makes similar triangles. The scale factor is the ratio of the distances from the light.
You can also work backward. If you are told the ratio of areas, take the square root to get $k$. For example, if one logo has area $40$ in.$^2$ and a similar one has area $360$ in.$^2$, then $\dfrac{360}{40} = 9$, so $k = \sqrt{9} = 3$.
A drone is flying over a park at a height of $15$ feet. Its camera projects a triangular field of view on the ground. If the drone climbs to a height of $45$ feet: a)…
A drone is flying over a park at a height of $15$ feet. Its camera projects a triangular field of view on the ground. If the drone climbs to a height of $45$ feet:
a) What is the scale factor between the original triangle of view and the new one?
b) If the perimeter of the area it covered on the ground was $120$ feet, what will the perimeter of the new area be?
c) If the surface (area) originally covered was $800\text{ ft}^2$, what will the surface covered at the new height be?
An outdoor cinema screen projects an image from a spotlight located $12$ feet away. The original screen has a surface area of $24$ $m^2$. If we move the screen closer to…
An outdoor cinema screen projects an image from a spotlight located $12$ feet away. The original screen has a surface area of $24$ $m^2$. If we move the screen closer to the projector until it is placed at exactly half the distance ($6$ feet from the spotlight), the projected image will be smaller but will keep the same shape. What will the area of the image be in this new position? Reason your answer using the relationship between the scale factor and the area.
Consider two right triangles. The first has legs of $3$ in. and $4$ in. The second triangle is similar to the first and its hypotenuse measures $15$ in. Calculate the…
Consider two right triangles. The first has legs of $3$ in. and $4$ in. The second triangle is similar to the first and its hypotenuse measures $15$ in. Calculate the perimeter and the area of this second triangle. (Hint: first find the hypotenuse of the original triangle to determine the scale factor).
Volume and putting it all together
For solids, there is one more step. If you scale a solid by $k$, its volume is multiplied by $k^3$, because length, width, and height all grow by $k$.
| Quantity | Multiplied by |
|---|---|
| Lengths, perimeter | $k$ |
| Area | $k^2$ |
| Volume | $k^3$ |
If a problem wants the answer in a different unit, convert after scaling. For example, in science and engineering you might see $1\text{ m}^3 = 1{,}000\text{ dm}^3$, or in customary units $1\text{ yd}^3 = 27\text{ ft}^3$.
Parallel segments inside triangles also lead to area questions. Look back at the triangle with $V$ at the top: the small triangle sits inside the big one, and the leftover region is a quadrilateral.
Two secants intersect at a point $V$. On one of them, points $A$ and $B$ are marked, and on the other, points $C$ and $D$, such that the segments $AC$ and $BD$ are…
Two secants intersect at a point $V$. On one of them, points $A$ and $B$ are marked, and on the other, points $C$ and $D$, such that the segments $AC$ and $BD$ are parallel (Thales's configuration). If we know that $VA = 9$ in., $VB = 15$ in., and the area of the smaller triangle $VAC$ is $27$ $cm^2$, calculate the area of the quadrilateral $ABDC$ given that the triangles are similar.
An architect presents a scale model of a cultural center in the shape of a triangular-based pyramid. The scale factor between the model and the actual building is…
An architect presents a scale model of a cultural center in the shape of a triangular-based pyramid. The scale factor between the model and the actual building is $1:150$. If the volume of the model is $0,8$ $dm^3$, what will the volume of the actual building be, expressed in cubic feet ($m^3$)? Remember to correctly apply the relationship between the scale factor and the volume.
The altitude of a right triangle
When you draw the altitude from the right angle down to the hypotenuse of a right triangle, you cut it into two smaller right triangles. Both are similar to the whole triangle, and to each other. That gives some handy formulas.
Why are the three triangles similar? All three are right triangles, and each small one shares an acute angle with the big one. By AA, they are similar. Compare the two small triangles: $\dfrac{p}{h} = \dfrac{h}{q}$. Compare the left small triangle with the whole: $\dfrac{p}{a} = \dfrac{a}{p+q}$. Cross-multiplying gives the formulas below.
A carpenter is designing a shelving unit reinforced with a crosspiece in the shape of a right triangle. In the base triangle, the height on the hypotenuse divides it…
A carpenter is designing a shelving unit reinforced with a crosspiece in the shape of a right triangle. In the base triangle, the height on the hypotenuse divides it into two segments measuring $18$ in. and $32$ in. The carpenter needs to know the length of the two battens that form the legs of the triangle in order to buy them. Calculate the length of both legs and the height of the crosspiece using the similarity relationships in right triangles.