Similar Triangles
Wilkin wants to know the height of a building in the Zona Colonial, but he can't climb it with a tape measure. With a mirror on the ground and a little geometry, he figures it out. What does it take to be sure that two triangles have the same shape? Less than you think: one of three conditions is enough, the postulates.
Congruence and similarity: postulates
Why are they enough? A triangle with three well-chosen pieces of data becomes "rigid." In SSS, three rods of fixed length can only be joined in one way. In SAS, the angle and the two sides fix the third vertex, and therefore the third side. In ASA, two rays with given angles leave the two ends of the side, and those rays meet at a single point: there is no other possible triangle.
Notice the connection: ASA is AA with one equal side ($k = 1$), and SSS and SAS are the same postulates with $k = 1$.
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 whether both triangles are similar, justifying your answer based on the similarity criteria.
Determine whether triangles $ABC$ and $DEF$ are similar, given that their sides measure $AB = 12\text{ cm}$, $BC = 16\text{ cm}$, $CA = 20\text{ cm}$ and…
Determine whether triangles $ABC$ and $DEF$ are similar, given 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.
The sides of a triangle measure $5\text{ cm}$, $7\text{ cm}$, and $9\text{ cm}$. Given 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 $9\text{ cm}$. Given 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.
Thales inside a triangle
When a line cuts two sides of a triangle parallel to the third side, two similar triangles appear.
Proof. Since $DE \parallel BC$, the angles $\angle ADE$ and $\angle ABC$ are corresponding angles between parallel lines, and therefore they are equal. The same holds for $\angle AED$ and $\angle ACB$. Angle $A$ is shared. By AA, $\triangle ADE \sim \triangle ABC$, and the corresponding sides are proportional: $\dfrac{AD}{AB} = \dfrac{AE}{AC} = \dfrac{DE}{BC}$. Also, $AB = AD + DB$ and $AC = AE + EC$, and subtracting 1 from each inverse ratio gives $\dfrac{AD}{DB} = \dfrac{AE}{EC}$.
Proof of the converse. If $\dfrac{AD}{AB} = \dfrac{AE}{AC}$, the triangles $ADE$ and $ABC$ share angle $A$ and the sides that form it are proportional. By SAS they are similar, so $\angle ADE = \angle ABC$. Those angles are corresponding angles, and if they are equal, the lines $DE$ and $BC$ are parallel.
In a triangle $ABC$, a segment $DE$ parallel to the base $BC$ is drawn such that point $D$ lies on side $AB$ and point $E$ on side $AC$. Given that $AD = 4\text{ cm}$,…
In a triangle $ABC$, a segment $DE$ parallel to the base $BC$ is drawn such that point $D$ lies on side $AB$ and point $E$ on side $AC$. Given 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 base $BC$.
You can do it on paper and send a photo with your answer.
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 $x$.
Shadows, mirrors, pinhole cameras, and crossed sightlines
Shadows, a mirror on the ground, a pinhole camera, or two sightlines that cross on the other side of a river create similar triangles by AA. This is Wilkin's method.
To calculate the height of a building, a student places a flat mirror on the ground at $12\text{ m}$ from the base of the building. Stepping back $1.5\text{ m}$ from the…
To calculate the height of a building, a student places a flat mirror on the ground at $12\text{ m}$ from the base of the building. Stepping back $1.5\text{ m}$ from the mirror, the student (whose eyes are at $1.6\text{ m}$ from the ground) can see exactly the top of the building reflected in it. Given that the angles of incidence and reflection at the mirror form similar right triangles, calculate the height of the building.
A pinhole camera (or camera obscura) has a depth of $15\text{ cm}$. It is placed in front of a building that is at a distance of $30\text{ m}$ from the pinhole. If the…
A pinhole camera (or camera obscura) has a depth of $15\text{ cm}$. It is placed in front of a building that is at a distance of $30\text{ m}$ from the pinhole. If the inverted image of the building projected onto 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. He walks…
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. He walks $15\text{ m}$ along the bank to a point $C$ and drives in a marker pole. He then walks $3\text{ m}$ further to point $D$, turns $90^\circ$ away from the river, and walks $4\text{ m}$ to a point $E$, from which he sees the marker pole $C$ and the tree $B$ aligned. Calculate the width $x$ of the river.
Right triangle: the altitude and leg theorems
In a right triangle $ABC$ (right angle at $C$), the altitude $h$ to the hypotenuse $c$ divides it into two projections $m$ and $n$ ($m + n = c$). Three similar triangles appear by AA: the large one and the two small ones, because each shares an acute angle with the large one and has a right angle.
Two theorems come out of the similarity:
Why? The two small triangles are similar to each other: $\dfrac{h}{m} = \dfrac{n}{h}$, so $h^2 = mn$. And the small triangle with the large one gives $\dfrac{a}{c} = \dfrac{m}{a}$, so $a^2 = mc$.
In a right-angled triangle, the altitude to the hypotenuse divides it into two segments (projections) measuring $4\text{ cm}$ and $9\text{ cm}$. Apply the geometric mean…
In a right-angled triangle, the altitude to the hypotenuse divides it into two segments (projections) measuring $4\text{ cm}$ and $9\text{ cm}$. Apply the geometric mean theorem (altitude theorem) to calculate the length of this altitude $h$.
In a right-angled triangle, the hypotenuse measures $10\text{ cm}$ and the projection of one of the legs onto it measures $3.6\text{ cm}$. Apply the cathetus theorem to…
In a right-angled triangle, the hypotenuse measures $10\text{ cm}$ and the projection of one of the legs onto it measures $3.6\text{ cm}$. Apply the cathetus theorem to find the length of this leg.
In a right-angled triangle, the altitude 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…
In a right-angled triangle, the altitude 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 theorems of similarity in right-angled triangles to calculate the perimeter and the area of the whole triangle.
In a right-angled 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…
In a right-angled 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 $6\text{ cm}$, analytically determine 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 cathetus theorem and the altitude theorem.
Triangles in a semicircle
If you take a diameter $AB$ of a circle and any point $C$ on it, the triangle $ABC$ is always a right triangle, with the right angle at $C$. This lets you use the altitude and leg theorems inside circles, spheres, and cones.
Why? Let $O$ be the center. Since $OA = OB = OC$ (radii), the triangles $OAC$ and $OBC$ are isosceles. Their base angles measure $\alpha$ (at $A$ and at $C$) and $\beta$ (at $B$ and at $C$). The three angles of $ABC$ add up to $\alpha + \beta + (\alpha + \beta) = 180^\circ$, so $\alpha + \beta = 90^\circ$, which is exactly the angle at $C$.
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-angled triangle formed by the…
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-angled triangle formed by the diameter of the sphere and the slant height of the cone to apply similarity theorems. Calculate the base radius of the cone and its height.
Areas and volumes with ratio k
If two similar figures have ratio $k$: lengths (sides, perimeters) are multiplied by $k$, areas by $k^2$, and volumes by $k^3$.
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?
A square-based pyramid has a height of $12\text{ cm}$ and the side length of its base is $8\text{ cm}$. The pyramid is cut by a plane parallel to the base, situated…
A square-based pyramid has a height of $12\text{ cm}$ and the side length of its base is $8\text{ cm}$. The pyramid is cut by a plane parallel to the base, situated $3\text{ cm}$ from the vertex. Calculate the area of the section obtained and the volume of the small top pyramid.
A water tank is in the shape of an inverted cone (vertex pointing downwards) with a top base radius of $3\text{ m}$ and a total depth of $4\text{ m}$. If the tank…
A water tank is in the shape of an inverted cone (vertex pointing downwards) with a top base radius 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.
Hidden similar triangles
Sometimes the triangles are not drawn: you have to find them inside an inscribed square, an extended trapezoid, or two poles with crossed cables.
In a right-angled 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…
In a right-angled 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 lies on the hypotenuse. Calculate the exact value of the side of the square $x$.
Two vertical poles of heights $4\text{ m}$ and $6\text{ m}$ stand on horizontal ground, separated by a distance of $10\text{ m}$. A straight cable is stretched from the…
Two vertical poles of heights $4\text{ m}$ and $6\text{ m}$ stand on horizontal ground, separated by a distance of $10\text{ m}$. A straight cable is stretched from the top of each pole to the base of the other. Calculate at what height $h$ above the ground the two cables cross. Prove whether or not this result depends on the distance separating the poles.
An isosceles trapezoid has bases measuring $15\text{ cm}$ and $25\text{ cm}$, and its height is $12\text{ cm}$. The non-parallel sides are extended until they intersect…
An isosceles trapezoid has bases measuring $15\text{ cm}$ 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 upper triangle and the total area of the large triangle formed.