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.

Marcos 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 $\hat{E} = 40^\circ$. Marcos 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 Marcos 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.

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 $k$.

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.

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.

During a cultural visit to the Plaza Mayor of Salamanca, 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.

During a route in the Hoces del Río Duratón Natural Park (Segovia), some environmental technicians need to find the width of the river between a willow tree located at point $A$ (on the opposite bank) and a marker placed at point $B$ (on their bank, directly opposite $A$). To measure it without crossing the water, they mark a straight line along their bank perpendicular to the segment $AB$ and drive in two stakes: one at point $C$, $24\text{ m}$ from $B$, and another at point $D$, $6\text{ m}$ from $C$ along the same line ($B$, $C$, and $D$ are aligned). From $D$, they walk perpendicularly to the bank away from the river to a point $E$ from which points $C$ and $A$ are seen perfectly aligned. The measured distance from $D$ to $E$ is $4,5\text{ m}$. Model the situation by identifying the two right-angled triangles, prove which criterion ensures they are similar, calculate the width of the river between $A$ and $B$, and determine between which values that width would vary if the measurement of $DE$ had a margin of error of $\pm 0,3\text{ m}$.

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.

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: Álvaro suggests checking whether $\dfrac{AD}{DB} = \dfrac{AE}{EC}$ by applying the converse of Thales's Theorem; Nuria 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.

An irrigators' community in Albacete manages a triangular plot $\triangle ABC$, whose base $BC$ measures $160\text{ m}$ and whose perpendicular height from vertex $A$ to said base measures $100\text{ m}$. A straight irrigation canal $MN$ parallel to the base $BC$ ($M$ on side $AB$ and $N$ on side $AC$) needs to be laid out to divide the estate into two parts of exactly equal area: an upper triangle $\triangle AMN$ and a lower trapezium $MNCB$. Justify which criterion guarantees that $\triangle AMN$ is similar to $\triangle ABC$, deduce the exact similarity ratio $k$ (in simplified radical form and to three decimal places) that divides the areas into $50\,\%$, calculate the length of the canal $MN$ and the distance from vertex $A$ at which it must be laid out. Finally, mathematically refute the intuitive proposal of a farmer who suggests digging the canal right at half the height ($50\text{ m}$), calculating what real percentage of the total area would remain in the upper triangle if his proposal were followed.