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.

Cian has to determine whether two triangles, $\triangle ABC$ and $\triangle DEF$, are similar. In triangle $\triangle ABC$, the sides measure $AB = 6\text{ cm}$, $AC = 9\text{ cm}$, and angle $\hat{B} = 40^\circ$. In triangle $\triangle DEF$, the sides measure $DE = 4\text{ cm}$, $DF = 6\text{ cm}$, and angle $\hat{E} = 40^\circ$. Cian states: "Since $\dfrac{AB}{DE} = \dfrac{AC}{DF} = \dfrac{6}{4} = \dfrac{9}{6} = 1.5$, the sides are proportional, and since both have an angle of $40^\circ$, they are similar by the SAS criterion". Identify and explain in detail the conceptual error Cian makes when applying the SAS criterion, and state which exact angle measurement you would need to know in both triangles to definitely guarantee their similarity.

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 $x$.

In a triangle $\triangle ABC$, the side lengths are $AB = 18\text{ cm}$ and $AC = 24\text{ cm}$. Two points are marked: $D$ on side $AB$ at $6\text{ cm}$ from $A$, and $E$ on side $AC$ at $8\text{ cm}$ from $A$. It is desired to prove whether segment $DE$ is parallel to side $BC$ and find what fraction of side $BC$ is measured by $DE$. To solve it, two strategies are proposed: Seán suggests checking whether $\dfrac{AD}{DB} = \dfrac{AE}{EC}$ by applying the converse of Thales's Theorem; Aoife suggests comparing triangles $\triangle ADE$ and $\triangle ABC$ directly using the SAS criterion sharing angle $\hat{A}$. Fully develop both methods with their justifications and critically analyse which of the two strategies is more complete for answering both questions, explaining the limitation of Thales's method regarding the ratio between the bases.

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-angled 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 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.

During a school trip to Dame Street in Dublin, a group of students decides to calculate the height of the City Hall façade on a clear day by means of shadow projection. A student who is $1.68\text{ m}$ tall casts at that moment a shadow of $1.20\text{ m}$ on the horizontal ground, while the building's shadow measures $25\text{ m}$. Before making exact calculations, estimate with justification whether the height of the building will be greater or less than the $25\text{ m}$ of its shadow, and propose an approximate mental range for that height. Next, prove which geometric criterion guarantees the similarity of the triangles formed, calculate the actual height of the building to two decimal places, and check whether your initial estimate was correct.

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?

You want to design a triangle $\triangle A'B'C'$ similar to a right-angled triangle $\triangle ABC$ whose legs measure $6\text{ cm}$ and $8\text{ cm}$. The set condition is that the area of the new triangle $\triangle A'B'C'$ must be exactly $96\text{ cm}^2$. First, calculate the area of the original triangle $\triangle ABC$ and deduce with justification the similarity ratio $k$ between the two. From that ratio, determine the exact length of the three sides of $\triangle A'B'C'$ and verify that the ratio between their perimeters also equals $k$.

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?

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.

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.

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.

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 altitude measures $6\text{ cm}$, analytically determine the length of the two segments and calculate the exact perimeter of the original triangle.

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.