Exercises: Similar triangles

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 $15\text{ cm}$, calculate the length of the other two sides of the second triangle.

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 $BC$.

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.

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 length of this leg.

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?

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.

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.

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.

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 $x$.

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.

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 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 $15\text{ m}$ 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 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.

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 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.

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.

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.

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.

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.

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 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.