Thales' Theorem

In the following drawing, lines $r$ and $s$ are intersected by three parallel lines ($a$, $b$, and $c$). Knowing that the segments formed on line $r$ measure $AB = 4\text{ cm}$ and $BC = 6\text{ cm}$, and that the corresponding segment to $AB$ on line $s$ measures $A'B' = 2\text{ cm}$, calculate the length of segment $B'C'$.

Calculate the value of the unknown $x$ in the following figure, knowing that the horizontal segments are parallel to each other.

In the following diagram, lines $r$ and $s$ are intersected by three parallel lines ($a$, $b$, and $c$). Use the given data to calculate the length of the unknown segment $x$.

In a design structure for a footbridge, three horizontal support beams are placed parallel to each other. The beams are joined by two inclined steel tie rods, as shown in the figure. Knowing that the segments formed on the first tie rod measure $x$ and $x + 2$ meters, and the corresponding segments on the second tie rod measure $3$ and $4$ meters, calculate the value of $x$ and the total length of the first tie rod.

A carpenter is building a shelving unit and wants to check if the two shelves are parallel. To do so, he measures the distances on the side supports. On the left side, the distance between the floor and the first shelf is $30\text{ cm}$, and between the first and second shelves it is $40\text{ cm}$. On the right side, the distance to the floor is $45\text{ cm}$ and between shelves it is $60\text{ cm}$. Are the shelves parallel? Justify your answer using the proportion from Thales' theorem.

In a designer shelving unit, three metallic supports that are parallel to each other are intersected by two wooden shelves. The top shelf is divided by the supports into two consecutive sections measuring $25$ cm and $x$. The bottom shelf is divided into two corresponding sections measuring $y$ and $48$ cm. Knowing that the top shelf measures $65$ cm in total, calculate the measurement of the sections marked with $x$ and $y$.

Find the fourth proportional of three segments whose lengths are $a = 8$ cm, $b = 12$ cm, and $c = 10$ cm. Perform both the numerical calculation and the geometric construction using Thales' theorem on two intersecting rays.

Check whether lines $r$ and $s$ in the following figure are parallel. To do so, verify whether the proportional relationship holds between the segments they form on the transversal lines intersecting them. Justify your answer with the necessary calculations.

Two triangles are in Thales' position. The small triangle has sides measuring $3\text{ cm}$, $4\text{ cm}$, and $5\text{ cm}$. If the side corresponding to the $3\text{ cm}$ side in the large triangle measures $9\text{ cm}$, how long will the other two sides of the larger triangle be?

A young graphic designer is creating a logo that includes two triangles in Thales' position. Knowing that segment $DE$ is parallel to $BC$, calculate the lengths of segments $AD$ and $BC$ given that $AB = 12$ cm, $AE = 5$ cm, $AC = 15$ cm, and $DE = 3$ cm.

A stepladder is resting against a wall. The rungs are parallel to the ground. The first rung is $30$ cm from the ground along the incline of the ladder, and the second is another $30$ cm from the first. If the horizontal distance from the base of the ladder to the wall is $1.2$ meters, what is the horizontal distance from the base to the vertical line of the second rung?

A carpenter is building a designer shelving unit in the shape of a right triangle. He has placed two horizontal shelves parallel to the base. The base of the shelving unit measures $1.2$ meters. The first shelf is located at a height of $40$ cm above the base and is $90$ cm long. The second shelf is $60$ cm long. At what height above the base is the second shelf located? What is the total height of the shelving unit? (Give all answers in centimeters).

Draw a segment $AB$ measuring $10\text{ cm}$ in your exercise book. Using Thales' theorem method, divide it into exactly $4$ equal parts. Briefly describe the steps you followed (drawing the auxiliary ray, making equal divisions, and joining the endpoints).

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

Divide a segment of $7\text{ cm}$ into two parts such that their ratio is $2$ to $3$. To do this, use an auxiliary ray and mark $5$ equal divisions ($2+3$). How long is each of the two new segments obtained on the original one?

Geometrically divide a segment $AB$ of length $9$ cm into three parts proportional to the numbers $2$, $3$, and $4$. Briefly explain the steps taken based on Thales' theorem.

A young person $1.5\text{ meters}$ tall casts a shadow $1\text{ metro}$ long. At that same moment, the shadow of a street lamp is $4\text{ meters}$ long. How tall is the street lamp? Draw a diagram with two right triangles to visualize the situation.

In a park, a tree $2\text{ meters}$ tall casts a shadow of $3.5\text{ meters}$. A few meters away, there is a statue whose shadow measures exactly $7\text{ meters}$. What is the height of the statue? Use the similarity of triangles established by Thales' theorem.

To measure the height of a basketball hoop, a Year 8 student places a vertical stick $1.5$ meters tall so that the end of its shadow coincides exactly with the end of the basketball hoop's shadow. If the stick's shadow is $2.1$ meters long and the distance from the base of the stick to the base of the basketball hoop is $2.2$ meters, how tall is the basketball hoop?

A street artist wants to project the shadow of a vertical sculpture $2.5$ meters tall onto a wall located at a certain distance. To do so, they place a spotlight on the ground $6$ meters from the sculpture. If the shadow projected on the wall is exactly $4$ meters tall, how far is the wall from the spotlight? If the artist moves the spotlight $2$ meters closer to the sculpture, how tall will the new shadow be?

In an adventure park, a zip line cable connects the top of two vertical towers. A safety bar $3$ meters high is placed perpendicular to the ground, so that it touches the zip line cable exactly. If the distance from the base of the small tower to the bar is $12$ meters and the distance from the bar to the cable's ground anchor is $8$ meters, calculate the height of the small tower. If the large tower is $30$ meters from the ground anchor, what is its height?

To measure the depth of a circular well with a diameter of $2$ meters, a person $1.70$ meters tall stands $1.2$ meters from the edge of the well. From that position, their eyes (located $1.60$ meters above the ground) see the edge of the well lined up with the bottom of the opposite wall. Draw a diagram of the situation and calculate the depth of the well using Thales' theorem.

Two buildings on the same street cast overlapping shadows at a given time of day. Building A is $18$ meters tall and casts a shadow of $12$ meters. Building B is located behind it such that its shadow ends at exactly the same point as Building A's shadow. If the distance from the base of Building A to the tip of its shadow is $12$ meters, and the distance between the two buildings is $10$ meters, calculate the height of Building B.

On a map with a scale of $1:2,500$, a piece of land in the shape of a right trapezoid has a longer base of $8$ cm and a shorter base of $5$ cm. If the actual height of this land is $100$ meters, calculate:

1. The actual length of the bases in meters.

2. The height on the map in centimeters.

3. If we draw a line parallel to the bases that divides the height on the map into two equal parts, how long will this new line be on the map?

Calculate the length of segments $y$ and $z$ in the following configuration of parallel lines cut by transversals. It is known that the ratio between the upper segments is $x / 2 = 5 / 4$. Use the data given in the image to set up the necessary proportions based on Thales' theorem.