Jayden's science teacher asks: "How tall is the flagpole in front of our school?" Nobody wants to climb it, and the tape measure only reaches 25 feet. Jayden is 5 ft tall, and on a sunny afternoon he casts a 6 ft shadow while the flagpole casts a 36 ft shadow. Is that enough information? With similar triangles, it is.
The three similarity criteria
Two figures are similar when they have the same shape but not necessarily the same size. One can be mapped onto the other by rigid motions and a dilation.
You do not have to check all six conditions. One of the following three 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.
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.
Once you know two triangles are similar, the scale factor $k = \dfrac{\text{new length}}{\text{original length}}$ gives you every missing side. If $k > 1$ the new triangle is bigger. If $k < 1$ it is smaller.
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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.
Triangles inside triangles: the Triangle Proportionality Theorem
Draw a line parallel to one side of a triangle so that it crosses the other two sides. It cuts off a small triangle that shares an angle with the big one. The other two angles are congruent because they are corresponding angles on parallel lines. That is AA, so the small triangle and the big triangle are similar.
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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 $x$.
Sunlight arrives as parallel rays, so at the same moment every vertical object makes the same angle with its shadow. Each object and its shadow form a right triangle. The two triangles share the sun's angle and a $90^\circ$ angle. That is AA.
Back to the opening question: the flagpole is 30 feet tall.
A streetlight works the same way, but the light is a single point, so the rays are not parallel. The lamp, the top of the person's head and the tip of the shadow lie on one line. That gives a small triangle inside a big one.
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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.
Shadows are not the only way to build similar triangles. A mirror on the ground, a sight line that crosses a river, and a pinhole camera all produce the same AA pair.
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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.
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.
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?
Draw the altitude from the right angle of a right triangle to the hypotenuse. It splits the triangle into two smaller right triangles. Both are similar to the original and to each other.
Why? In $\triangle ABC$ the angle at $C$ is $90^\circ$. $\triangle ACD$ has a right angle at $D$ and shares $\angle A$ with it. $\triangle CBD$ has a right angle at $D$ and shares $\angle B$. By AA,
Reading the proportions from these similarities gives two famous results. A number $x$ is the geometric mean of $m$ and $n$ when $x^2 = mn$, that is, $x = \sqrt{mn}$.
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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.
Many problems do not show a right triangle with its altitude. You have to spot one. Here are three places to look:
Rectangle: the diagonal cuts it into two right triangles. The perpendicular from a vertex to the diagonal is the altitude to the hypotenuse, and it splits the diagonal into two projections.
Semicircle: an angle inscribed in a semicircle is a right angle. If $AB$ is a diameter and $C$ is on the circle, then $\triangle ABC$ is right-angled at $C$.
Cone inside a sphere: if the apex $A$ and the opposite pole $B$ of the sphere form a diameter, and $C$ is a point on the rim of the cone's base, then $\angle ACB = 90^\circ$. The slant height $AC$ is a leg, the base radius is the altitude from $C$ to $AB$, and the projection of $AC$ on $AB$ is the cone's height.
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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.
If every length in a figure is multiplied by $k$, the perimeter is a sum of lengths, so it is multiplied by $k$. The area depends on two lengths (base and height), so it is multiplied by $k \cdot k$.
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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?
Cut a pyramid or a cone with a plane parallel to its base. The cut face is similar to the base. The small pyramid or cone above the cut is similar to the whole solid. The scale factor is the ratio of the distances from the apex.
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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.
The hardest problems do not draw the two triangles for you. Look for parallel lines (AA by corresponding angles), shared angles, and right angles. Then label which triangle each length belongs to, small or big, before you write a proportion.
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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.