Exercises: Similar triangles

Check whether a right-angled triangle with an acute angle of $35^\circ$ is similar to another right-angled triangle with an acute angle of $55^\circ$. Justify your answer using similarity tests.

The sides of a triangle measure $5 \text{ cm}$, $7 \text{ cm}$, and $9 \text{ cm}$. A similar triangle is constructed whose shortest side measures $15 \text{ cm}$. Calculate the lengths of the other two sides of the new triangle.

In the given diagram, line segment $BC$ is parallel to $DE$. Given that $AB = 6 \text{ cm}$, $BC = 8 \text{ cm}$, and $DE = 12 \text{ cm}$, calculate the length of segment $BD$ (marked as $x$).

The perimeter of a triangle $T_1$ is $24 \text{ cm}$. A triangle $T_2$ is constructed similar to $T_1$ with a scale factor of $k = 3$. What is the perimeter of triangle $T_2$?

The area of a triangle is $10 \text{ cm}^2$. If we draw a similar triangle in which each side is twice as long as the sides of the original, what will be the area of the new triangle?

A vertical pole with a height of $2 \text{ m}$ casts a shadow of $1,5 \text{ m}$ on flat ground. At the same moment, a nearby building casts a shadow of $12 \text{ m}$. Calculate the height of the building.

The diagram illustrates the situation described:

In the right-angled triangle in the diagram, the height relative to the hypotenuse divides it into two segments of lengths $4 \text{ cm}$ and $9 \text{ cm}$. Calculate the value of this height $h$.

In a right-angled triangle, the hypotenuse measures $10 \text{ cm}$ and the projection of one of the legs onto the hypotenuse measures $3,6 \text{ cm}$. Apply the leg theorem to calculate the length of that leg.

Hugo claims that two set squares are similar. The first has sides of $6\text{ cm}$, $8\text{ cm}$, and $10\text{ cm}$. The second has sides of $9\text{ cm}$, $12\text{ cm}$, and $16\text{ cm}$. To justify this, Hugo writes: “They are similar because $\dfrac{9}{6} = 1,5$ and $\dfrac{12}{8} = 1,5$; since two sides are already proportional, there is no need to check the third side.”\na) Identify the error in Hugo's reasoning and explain what condition is missing to ensure similarity without measuring the third side.\nb) Correct his conclusion: check the proportion with the three corresponding sides and determine whether the two set squares are really similar.

During a cultural visit to Sevilla, a group of 4th-year ESO students wants to find out the height of the Torre del Oro without using the internet. To do this, they take advantage of a clear day: Irene, who is $1,60\text{ m}$ tall, stands on the esplanade next to the Guadalquivir River and her classmates measure her shadow, which is $1,20\text{ m}$. At that same moment, they measure the shadow cast by the Torre del Oro, which reaches $27\text{ m}$.\na) Justify why the triangle formed by Irene and her shadow is similar to the one formed by the tower and its shadow, indicating the applicable similarity criterion.\nb) Model the situation using a geometric proportion and calculate the actual height of the Torre del Oro.

To calculate the height of a building, a student places a mirror on the ground $15 \text{ m}$ from the base of the building. They step back $2 \text{ m}$ until, standing upright, they can see the top of the building reflected in the mirror. If the student's eyes are $1,6 \text{ m}$ above the ground, calculate the height $h$ of the building.

Two surveyors want to measure the width of a river (distance $d$ between parallel banks). They drive pegs into their bank $20 \text{ m}$ apart. On the opposite bank, they identify two trees $40 \text{ m}$ apart. They trace crossed sightlines that intersect at a point $P$. If the perpendicular distance from $P$ to the surveyors' bank is $15 \text{ m}$, what is the total width $d$ of the river? (Note: assume that the segments $20 \text{ m}$ and $40 \text{ m}$ are parallel to each other).

A triangular field has an area of $450 \text{ m}^2$ and its base measures $30 \text{ m}$. A plan of the field is to be drawn in which the area of the drawn triangle is $18 \text{ cm}^2$. How long will the base of the triangle be on the plan?

A person $1,70 \text{ m}$ tall walks away from a lamppost. When they are $3 \text{ m}$ from the base of the lamppost, their shadow cast on the ground measures $2 \text{ m}$. Calculate the total height of the lamppost.

The diagram summarises the given data:

The legs of a right-angled triangle measure $15 \text{ cm}$ and $20 \text{ cm}$. Calculate the length of the projections of both legs onto the hypotenuse using similarity relationships or theorems derived from them.

A company logo is an isosceles triangle whose sides measure $10 \text{ cm}$, $10 \text{ cm}$, and $12 \text{ cm}$ (base). For design reasons, a line parallel to the base is drawn at a distance of $3 \text{ cm}$ from the opposite vertex. Calculate the area of the small triangle formed at the top.

You can do it on paper and send a photo with your answer.

In a right-angled triangle, the height to the hypotenuse is $4,8 \text{ cm}$ and the projection of the shorter leg onto the hypotenuse is $3,6 \text{ cm}$. Calculate the full perimeter of the original right-angled triangle.

A camera has its sensor at a distance of $50 \text{ mm}$ from the optical centre of the lens. If it focuses on a tree that has a height of $4 \text{ m}$ and the inverted image projected onto the sensor has a height of $10 \text{ mm}$, how far from the lens is the tree?

In a wind farm in Albacete, a maintenance technician needs to know the total height of a wind turbine. Since she cannot measure it directly, she drives a vertical $2\text{ m}$ high rod into the ground. At that moment, the rod's shadow measures $1,5\text{ m}$ and the shadow of the wind turbine on the flat ground measures $78\text{ m}$.\na) Before calculating, estimate the result mentally: will the height of the wind turbine be greater or less than $100\text{ m}$? Justify your estimate by comparing the height of the rod with the length of its shadow.\nb) Model the situation using triangle similarity and calculate the exact height of the wind turbine.\nc) Check whether the exact result obtained is consistent with your initial estimate and assess whether it makes sense for the typical dimensions of a wind turbine in Spain.

A road signage company manufactures warning signs in the shape of a right triangle. The standard model has legs of $30\text{ cm}$ and $40\text{ cm}$ (and a hypotenuse of $50\text{ cm}$). For a highway, the Directorate-General for Traffic (DGT) requests an enlarged version that is similar to the standard one, but meets a specific requirement: its usable surface area must be exactly $2400\text{ cm}^2$.\na) Calculate the area of the standard model and identify the mathematical relationship between the linear scale factor $k$ and the ratio of the areas of two similar triangles.\nb) From the previous result, determine the value of $k$ and deduce the exact lengths that the three sides of the highway sign must have.\nc) If they also request the manufacturing of a smaller similar version whose total perimeter is exactly $60\text{ cm}$, determine the scale factor with respect to the standard model and the measurements of its three sides.

In a right-angled triangle with legs $8 \text{ cm}$ and $6 \text{ cm}$, a square of side $x$ is inscribed such that one vertex of the square coincides with the vertex of the right angle, and the opposite vertex lies on the hypotenuse. Calculate the side length $x$ of the square using similar triangles.

Two vertical poles of heights $4 \text{ m}$ and $6 \text{ m}$ are separated by a distance of $10 \text{ m}$ on flat ground. Two taut cables are stretched: one from the top of the first pole to the base of the second, and another from the top of the second pole to the base of the first. At what height $h$ above the ground do the two cables cross? Does the result depend on the distance $10 \text{ m}$?

In a triangle $ABC$, a line segment $MN$ is drawn parallel to side $BC$, with $M$ on $AB$ and $N$ on $AC$. The lengths of the segments formed are: $AM = x$, $MB = x+2$, $AN = x+1$, and $NC = x+4$. Determine the value of $x$ and the total lengths of sides $AB$ and $AC$.

You can do it on paper and send a photo with your answer.

A tank has the shape of a right circular cone. The radius of its base measures $3 \text{ m}$ and its height is $4 \text{ m}$. A right cylinder resting on the base of the cone is to be inscribed inside it. If the radius of the cylinder is $r$, express the height $h$ of the cylinder in terms of $r$ using similar triangles.

In a trapezium $ABCD$ with bases $AB = 10 \text{ cm}$ and $CD = 4 \text{ cm}$, the diagonals $AC$ and $BD$ intersect at point $O$. If the area of triangle $COD$ is $12 \text{ cm}^2$, calculate the area of triangle $AOB$ and prove your reasoning.

A triangle of area $S$ is cut by two lines parallel to its base such that the height of the original triangle is divided into three equal parts. This divides the shape into a small top triangle, a central trapezium, and a bottom trapezium. Determine what fraction of the total area $S$ corresponds to the central trapezium.

Prove, using the leg theorem and the altitude theorem (or the area of a triangle), that in any right-angled triangle with legs $b$ and $c$, and height to the hypotenuse $h$, the following relationship holds:

A driver is sitting in his car. His eyes are $1,2 \text{ m}$ above the ground. The nose of the car is horizontal, with a length of $1,5 \text{ m}$ (horizontal distance from his eyes to the front end) and a height of $0,8 \text{ m}$. Calculate the minimum distance $d$ in front of the car at which an object on the ground must be for the driver to see it (that is, the length of the front "blind spot").

To roof an industrial warehouse, an inclined beam is installed, forming a right triangle with the ground and the wall. To reinforce the structure, a vertical strut of height $h$ perpendicular to the inclined beam is placed (that is, the altitude to the hypotenuse), dividing the beam into two segments of lengths $m = 1,8\text{ m}$ and $n = 3,2\text{ m}$. Two members of the technical team propose different methods to find the length $h$ of the strut:\n- Method 1 (Álvaro): Apply the Pythagorean theorem to the two smaller right triangles and to the total triangle to solve a system of three equations.\n- Method 2 (Beatriz): Directly use the relationship $h^2 = m \cdot n$ obtained from triangle similarity.\na) Calculate the height $h$ of the strut using Method 2.\nb) Compare both methods and justify why Beatriz's method is much more efficient. Also explain where Beatriz's formula comes from mathematically: which two right triangles are being compared and by which criterion are they similar?\nc) If you also need to calculate the length of the beam segment $b$ resting on segment $m$, what direct formula derived from similarity (geometric mean theorem / leg rule) allows you to find it without solving any system?

An $18\text{ m}$ high residential building in Zaragoza is located across from a high school yard. The yard has a sports court that starts $20\text{ m}$ away from the building's facade and ends $35\text{ m}$ away. The management team wants to analyze the impact of the building's shadow on this court:\na) At midday in December (winter solstice), a vertical traffic sign of height $2,5\text{ m}$ located next to the building casts a shadow of $3,5\text{ m}$. Model the situation using similar triangles, calculate the length of the building's shadow, and determine with reasoning whether it covers the sports court (completely, partially, or not at all).\nb) At midday in June (summer solstice), the Sun is higher and the same $2,5\text{ m}$ sign casts a shadow of only $0,9\text{ m}$. Without needing to calculate the exact numerical value of the building's shadow step by step, reason from the constant of proportionality whether the building's shadow will reach the sports court in summer.