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 whether 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 $9\text{ cm}$. 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.

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 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 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 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.

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.

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-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.

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 (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 $15\text{ m}$ 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.

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.

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 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.

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.

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.

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.

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.

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.