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.

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.

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 $EC = 3\text{ cm}$. Apply Thales's theorem to set up an equation and calculate the numerical value of $x$.

To calculate the height of a building, a student places a plane 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-angled 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. 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.

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 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 $3\text{ cm}$ 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.