Parallel Lines and Proportional Segments

Calculate the value of the unknown $x$ in the following figure, given 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 data provided to calculate the length of the unknown segment $x$.

In a design structure for a pedestrian bridge, three horizontal reinforcing beams have been placed parallel to each other. The beams are joined by two oblique steel tie rods, as shown in the figure. Knowing that the segments determined on the first tie rod measure $x$ and $x + 2$ feet, and the corresponding segments on the second tie rod measure $3$ and $4$ feet, calculate the value of $x$ and the total length of the first tie rod.

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

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

A carpenter is building a designer shelving unit in the shape of a right triangle. They have placed two horizontal shelves parallel to the base. The base of the shelving unit measures $1,2$ feet. The first shelf is located at a height of $40$ in. from the base and is $90$ in. long. The second shelf is $60$ in. long. At what height from the base is the second shelf located? What is the total height of the shelving unit? (Express all results in inches).

A carpenter is building a shelving unit and wants to check if the two shelves are parallel. To do this, 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 there is $40\text{ cm}$. On the right side, the distance to the floor is $45\text{ cm}$ and between shelves there is $60\text{ cm}$. Are the shelves parallel? Justify your answer using the proportion from Thales's theorem.

Check if lines $r$ and $s$ in the following figure are parallel. To do this, verify if the proportionality relationship is met between the segments they determine on the intersecting transversals. Justify your answer with the necessary calculations.

A young man of height $1,5\text{ meters}$ casts a shadow $1\text{ metro}$ long. At that same moment, the shadow of a street lamp measures $4\text{ meters}$. What is the height of the street lamp? Draw a diagram with two right triangles to visualize the situation.

In a park, a tree of height $2\text{ meters}$ casts a shadow of $3,5\text{ meters}$. A few feet 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's theorem.

To measure the height of a basketball hoop, an 8th grade student places a vertical stick $1,5$ feet high so that the end of its shadow exactly coincides with the end of the hoop's shadow. If the stick's shadow measures $2,1$ feet and the distance from the base of the stick to the base of the hoop is $2,2$ feet, what is the height of the hoop?

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

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 ($2+3$). What is the length of each of the two new segments obtained on the original one?

Graphically divide a segment $AB$ of length $9$ in. into three parts that are proportional to the numbers $2$, $3$ and $4$. Briefly explain the steps taken based on Thales's Theorem.

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

A stepladder is leaning against a wall. The rungs are parallel to the ground. The first rung is $30$ in. from the ground following the inclination of the ladder, and the second is another $30$ in. from the first. If the horizontal distance from the base of the ladder to the wall is $1,2$ feet, what is the horizontal distance from the base to the vertical of the second rung?

Two buildings located on the same street cast shadows that overlap at a given time of day. Building A has a height of $18$ feet and casts a shadow of $12$ feet. Building B is located behind it so that its shadow ends at exactly the same point as that of building A. If the distance between the base of building A and the end of its shadow is $12$ feet, and the distance between the two buildings is $10$ feet, calculate the height of building B.

To measure the depth of a circular well $2$ feet in diameter, a person $1,70$ feet tall stands $1,2$ feet from the edge of the well. From that position, their eyes (located $1,60$ feet from the ground) see the edge of the well aligned with the bottom end of the opposite wall. Draw a diagram of the situation and calculate the depth of the well using Thales's Theorem.

On a map with a scale of $1:2.500$, a plot of land in the shape of a right trapezoid has a larger base of $8$ in. and a smaller base of $5$ in. If in reality the height of this land is $100$ feet, calculate:

1. The actual length of the bases in feet.

2. The measurement of the height on the map in inches.

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