How tall is the flagpole in front of your school? Nobody wants to climb it with a tape measure. But you can measure your own shadow, the pole's shadow, and your own height, and that is enough. The secret is similar triangles: triangles with the same shape, so all their lengths are in the same ratio.
Three ways to prove triangles are similar
Order matters: in $\triangle ABC \sim \triangle DEF$, vertex $A$ matches $D$, $B$ matches $E$, and $C$ matches $F$. The scale factor works for every pair of corresponding lengths: sides, altitudes, medians, perimeters.
You don't have to check everything. One of these three tests is enough.
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Two triangles have, respectively, the following angles: the first has an angle of $45^\circ$ and another of $60^\circ$; the second has an angle of $60^\circ$ and another…
Two triangles have, respectively, the following angles: the first has an angle of $45^\circ$ and another of $60^\circ$; the second has an angle of $60^\circ$ and another of $75^\circ$. Determine if both triangles are similar, justifying your answer based on the similarity criteria.
The sides of a triangle measure $5\text{ cm}$, $7\text{ cm}$ and $9\text{ cm}$. Knowing that a second triangle is similar to this one and that its shortest side measures…
The sides of a triangle measure $5\text{ cm}$, $7\text{ cm}$ and $9\text{ cm}$. Knowing that a second triangle is similar to this one and that its shortest side measures $15\text{ cm}$, calculate the length of the other two sides of the second triangle.
Determine if the triangles $ABC$ and $DEF$ are similar, knowing that their sides measure $AB = 12\text{ cm}$, $BC = 16\text{ cm}$, $CA = 20\text{ cm}$ and…
Determine if the triangles $ABC$ and $DEF$ are similar, knowing that their sides measure $AB = 12\text{ cm}$, $BC = 16\text{ cm}$, $CA = 20\text{ cm}$ and $DE = 9\text{ cm}$, $EF = 12\text{ cm}$, $FD = 15\text{ cm}$. If so, state the criterion used and the scale factor.
Draw a line parallel to one side of a triangle so that it cuts the other two sides. Two triangles appear, one nested inside the other. They share the angle at $A$, and the other angles are congruent because they are corresponding angles formed by parallel lines. That is AA.
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In a triangle $ABC$, a segment $DE$ is drawn parallel to the base $BC$, such that point $D$ belongs to side $AB$ and point $E$ to side $AC$. Knowing that…
In a triangle $ABC$, a segment $DE$ is drawn parallel to the base $BC$, such that point $D$ belongs to side $AB$ and point $E$ to side $AC$. Knowing that $AD = 4\text{ cm}$, $DB = 6\text{ cm}$ and $DE = 5\text{ cm}$, apply Thales's theorem in the triangle to calculate the length of the base $BC$.
In the following figure, the segments $DE$ and $BC$ are parallel. The lengths are given by algebraic expressions: $AD = x$, $DB = 2\text{ cm}$, $AE = x + 2$, and…
In the following figure, the segments $DE$ and $BC$ are parallel. The lengths are given by algebraic expressions: $AD = x$, $DB = 2\text{ cm}$, $AE = x + 2$, and $EC = 3\text{ cm}$. Apply Thales's theorem to set up an equation and calculate the numerical value of $x$.
At the same moment, the sun's rays are parallel. A vertical object and its shadow form a right triangle, and every vertical object makes the same angle with its ray. So two objects and their shadows give similar right triangles (AA).
A lamp works the same way, but now the light rays all start at one point. Look for the big triangle (lamp to the tip of the shadow) and the small triangle (person to the tip of the shadow).
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A tree casts a shadow of $8\text{ m}$ at the same instant that a vertical post $1,5\text{ m}$ high casts a shadow of $2\text{ m}$. Mentally draw the right triangles that…
A tree casts a shadow of $8\text{ m}$ at the same instant that a vertical post $1,5\text{ m}$ high casts a shadow of $2\text{ m}$. Mentally draw the right triangles that are formed and calculate the height of the tree.
A streetlight has a height of $4,5\text{ m}$. A person who is $1,8\text{ m}$ tall walks away from the streetlight in a straight line. At a given moment, the person is…
A streetlight has a height of $4,5\text{ m}$. A person who is $1,8\text{ m}$ tall walks away from the streetlight in a straight line. At a given moment, the person is $3\text{ m}$ from the base of the streetlight. Calculate the length $x$ of the shadow cast by the person at that instant.
A spotlight is on the ground $12\text{ m}$ from a vertical wall. A person $1,8\text{ m}$ tall walks from the spotlight towards the wall at a constant speed of…
A spotlight is on the ground $12\text{ m}$ from a vertical wall. A person $1,8\text{ m}$ tall walks from the spotlight towards the wall at a constant speed of $1\text{ m/s}$. Apply the similarity of triangles to calculate the height $y$ of the person's shadow projected on the wall when $4\text{ s}$ have elapsed since they started walking.
You don't need the sun. Any setup where angles match gives similar triangles.
Mirror. Light bounces off a flat mirror so that the angle of incidence equals the angle of reflection. Put a mirror on the ground, stand where you can see the top of the building in it, and the two triangles (you and the building) are similar by AA.
Across a river. Stand at $A$ directly across from a tree $B$. Walk along the bank to $C$, then to $D$, turn and walk away from the river to $E$ until $C$, $E$, and $B$ line up. Triangles $ABC$ and $DEC$ have right angles at $A$ and $D$ and vertical angles at $C$, so they are similar by AA.
Pinhole camera. Light from the top of an object passes straight through the tiny hole and lands upside down on the back wall. The two triangles (object-to-hole and hole-to-image) meet at the hole with vertical angles, so they are similar. The image is inverted, but the ratio is what matters.
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To calculate the height of a building, a student places a flat mirror on the ground $12\text{ m}$ from the base of the building. By stepping back $1,5\text{ m}$ from the…
To calculate the height of a building, a student places a flat mirror on the ground $12\text{ m}$ from the base of the building. By stepping back $1,5\text{ m}$ from the mirror, the student (whose eyes are $1,6\text{ m}$ from the ground) can see exactly the highest part of the building reflected in it. Knowing that the angles of incidence and reflection in the mirror form similar right triangles, calculate the height of the building.
A pinhole camera has a depth of $15\text{ cm}$. It is placed in front of a building that is $30\text{ m}$ away from the camera's pinhole. If the inverted image of the…
A pinhole camera has a depth of $15\text{ cm}$. It is placed in front of a building that is $30\text{ m}$ away from the camera's pinhole. If the inverted image of the building projected on the back of the camera has a height of $12\text{ cm}$, what is the actual height of the building?
To measure the width of a river with parallel banks, a surveyor stands at a point $A$ on one bank, directly opposite a tree $B$ on the opposite bank. They walk…
To measure the width of a river with parallel banks, a surveyor stands at a point $A$ on one bank, directly opposite a tree $B$ on the opposite bank. They walk $15\text{ m}$ along the bank to a point $C$ and drive in a ranging pole. Then they walk $3\text{ m}$ further to point $D$, turn $90^\circ$ away from the river and walk $4\text{ m}$ to a point $E$, from which they see the ranging pole $C$ and the tree $B$ aligned. Calculate the width $x$ of the river.
If two figures are similar with scale factor $k$, lengths are not the only thing that changes. Area changes faster than length, because area has two dimensions.
You can also go backward. If two similar figures have areas $18$ and $50$, the area ratio is $\dfrac{50}{18} = \dfrac{25}{9}$, so $k = \sqrt{\dfrac{25}{9}} = \dfrac{5}{3}$.
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The scale factor between two similar polygons is $k = 2,5$. If the area of the smaller polygon is $12\text{ cm}^2$, what is the area of the larger polygon?
The scale factor between two similar polygons is $k = 2,5$. If the area of the smaller polygon is $12\text{ cm}^2$, what is the area of the larger polygon?
Two companies manufacture similar triangular sails for racing boats. The perimeter of the small sail is $12\text{ m}$ and that of the large sail is $18\text{ m}$. If the…
Two companies manufacture similar triangular sails for racing boats. The perimeter of the small sail is $12\text{ m}$ and that of the large sail is $18\text{ m}$. If the area of the small sail is $6\text{ m}^2$, what is the area of the large sail?
Slice a pyramid or a cone with a plane parallel to the base. The slice is similar to the base, and the small piece on top is a smaller copy of the whole solid. The scale factor is
$$k = \frac{\text{distance from the vertex to the slice}}{\text{total height}}$$
The same idea works for liquids. A cone-shaped cup with water in it contains a smaller cone of water, similar to the cup.
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A square-based pyramid has a height of $12\text{ cm}$ and the side of its base measures $8\text{ cm}$. The pyramid is cut by a plane parallel to the base, located…
A square-based pyramid has a height of $12\text{ cm}$ and the side of its base measures $8\text{ cm}$. The pyramid is cut by a plane parallel to the base, located $3\text{ cm}$ from the vertex. Calculate the area of the cross-section obtained and the volume of the small upper pyramid.
A water tank has the shape of an inverted cone (vertex pointing downwards) with a radius at the top base of $3\text{ m}$ and a total depth of $4\text{ m}$. If the tank…
A water tank has the shape of an inverted cone (vertex pointing downwards) with a radius at the top base of $3\text{ m}$ and a total depth of $4\text{ m}$. If the tank contains water up to a height of $2\text{ m}$, use the relationship between the volumes of similar solids to calculate the volume of water it contains and the remaining empty volume.
Draw the altitude from the right angle of a right triangle down to the hypotenuse. It splits the triangle into two smaller right triangles. Each shares an acute angle with the big one, so by AA all three triangles are similar.
From $\triangle ACD \sim \triangle CBD \sim \triangle ABC$, writing the proportions of corresponding sides gives:
Some problems call the second theorem simply the leg theorem.
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In a right triangle, the altitude drawn to the hypotenuse divides it into two segments (projections) that measure $4\text{ cm}$ and $9\text{ cm}$. Apply the geometric…
In a right triangle, the altitude drawn to the hypotenuse divides it into two segments (projections) that measure $4\text{ cm}$ and $9\text{ cm}$. Apply the geometric mean altitude theorem to calculate the length of this altitude $h$.
In a right triangle, the hypotenuse measures $10\text{ cm}$ and the projection of one of the legs onto it measures $3,6\text{ cm}$. Apply the leg theorem to find the…
In a right triangle, the hypotenuse measures $10\text{ cm}$ and the projection of one of the legs onto it measures $3,6\text{ cm}$. Apply the leg theorem to find the length of this leg.
In a right triangle, the altitude relative to the hypotenuse measures $6\text{ cm}$ and the projection of one of the legs onto the hypotenuse is $4,5\text{ cm}$. Apply…
In a right triangle, the altitude relative to the hypotenuse measures $6\text{ cm}$ and the projection of one of the legs onto the hypotenuse is $4,5\text{ cm}$. Apply the similarity theorems in right triangles to calculate the perimeter and the area of the complete triangle.
Right triangles with unknowns, rectangles, and spheres
Sometimes the right triangle is hidden, or the unknown is a segment. Then an algebra equation (often a quadratic) does the work.
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In a right triangle, the altitude to the hypotenuse divides it into two segments (projections) such that one is $5\text{ cm}$ longer than the other. If the altitude…
In a right triangle, the altitude to the hypotenuse divides it into two segments (projections) such that one is $5\text{ cm}$ longer than the other. If the altitude measures $6\text{ cm}$, determine analytically the length of the two segments and calculate the exact perimeter of the original triangle.
The diagonal of a rectangle measures $20\text{ cm}$. If we draw the perpendicular from one of the vertices to the diagonal, it is divided into two segments. If one of…
The diagonal of a rectangle measures $20\text{ cm}$. If we draw the perpendicular from one of the vertices to the diagonal, it is divided into two segments. If one of the segments measures $7,2\text{ cm}$, find the total area of the rectangle by applying the leg and altitude theorems.
A right cone whose slant height measures $g = 24\text{ cm}$ is inscribed in a sphere of radius $R = 15\text{ cm}$. Consider the right triangle formed by the diameter of…
A right cone whose slant height measures $g = 24\text{ cm}$ is inscribed in a sphere of radius $R = 15\text{ cm}$. Consider the right triangle formed by the diameter of the sphere and the slant height of the cone to apply the similarity theorems. Calculate the radius of the base of the cone and its height.
In harder problems, the two triangles aren't drawn. Look for parallel lines (equal corresponding angles), right angles, shared angles, and vertical angles. Then name the triangles in matching order and set up one proportion.
A square in a right triangle. A right triangle has legs $12$ ft and $4$ ft. A square of side $x$ has one vertex at the right angle and the opposite vertex on the hypotenuse.
Crossing cables. Two vertical posts stand on level ground. A cable runs from the top of each post to the base of the other.
Extending a trapezoid. Extend the slanted sides of a trapezoid until they meet. The small triangle on top is similar to the big triangle, because the bases are parallel.
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In a right triangle with legs $6\text{ cm}$ and $8\text{ cm}$, a square of side $x$ is inscribed such that one of the vertices of the square coincides with the vertex of…
In a right triangle with legs $6\text{ cm}$ and $8\text{ cm}$, a square of side $x$ is inscribed such that one of the vertices of the square coincides with the vertex of the right angle of the triangle, and the opposite vertex rests on the hypotenuse. Calculate the exact value of the side of the square $x$.
Two vertical posts of heights $4\text{ m}$ and $6\text{ m}$ are situated on horizontal ground, separated by a distance of $10\text{ m}$. A straight cable is stretched…
Two vertical posts of heights $4\text{ m}$ and $6\text{ m}$ are situated on horizontal ground, separated by a distance of $10\text{ m}$. A straight cable is stretched from the top of each post to the base of the other. Calculate at what height $h$ from the ground both cables cross. Show whether or not this result depends on the distance separating the posts.
Consider a right trapezoid whose bases measure $15\text{ cm}$ and $25\text{ cm}$, and its height is $12\text{ cm}$. The non-parallel sides are extended until they…
Consider a right trapezoid whose bases measure $15\text{ cm}$ and $25\text{ cm}$, and its height is $12\text{ cm}$. The non-parallel sides are extended until they intersect at a point, forming a triangle at the top. Calculate the height $x$ of this top triangle and the total area of the large triangle formed.
Back to the start: with a $6$ ft student casting a $4$ ft shadow and the flagpole's shadow measuring $30$ ft, the similar triangles give a flagpole of $45$ ft, with nobody climbing anything.