Theorem 12 (parallel lines and proportion)

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 $B'C'$.

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 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 $x$.

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.

It is planned to design a triangular reinforcing structure $ABC$ for a crane, where the horizontal base $BC$ measures $120\text{ cm}$ and the inclined side $AB$ measures $90\text{ cm}$. A horizontal steel tie rod $DE$, parallel to base $BC$, must be welded, with point $D$ located on side $AB$ and point $E$ on side $AC$.

1. a) If the length of the tie rod $DE$ needs to be exactly $40\text{ cm}$, at what distance from vertex $A$ must point $D$ be welded along side $AB$?
2. b) If it is decided to modify the design so that the tie rod $DE$ divides side $AB$ such that the upper segment $AD$ measures exactly twice as much as the lower segment $DB$, how long will the bar $DE$ be in this new case?
3. c) Deduce an algebraic formula that expresses the length of the horizontal bar $DE$ (which we will call $y$) in terms of the distance $AD$ (which we will call $x$), and indicate between which values it makes geometric sense for $x$ to vary.

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.

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.

A young person $1.5\text{ metres}$ tall casts a shadow $1\text{ metre}$ 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?