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 $B'C'$.

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 $x$.

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.

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 $1,5\text{ metres}$ tall casts a shadow $1\text{ metro}$ long. At that same moment, the shadow of a street lamp is $4\text{ metres}$ long. How tall is the street lamp? Draw a diagram with two right-angled triangles to visualise the situation.

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

In a park, a tree $2\text{ metres}$ tall casts a shadow of $3,5\text{ metres}$. A few metres away, there is a statue whose shadow measures exactly $7\text{ metres}$. 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 $1,5$ metres 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 $2,1$ metres long and the distance from the base of the stick to the base of the basketball hoop is $2,2$ metres, how tall is the basketball hoop?