Exercises: 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 $B'C'$.

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.

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.

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 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.

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

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?

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.

In your high school playground, at $12:00$, you stick a $1\text{ m}$ tall ruler vertically into the ground and observe that its shadow measures exactly $0.5\text{ m}$. Right next to it, at that very same moment, a lamppost casts a shadow of $2.6\text{ m}$.

1. a) Without carrying out pen-and-paper calculations, estimate in a reasoned manner whether the lamppost will measure more or less than $5\text{ m}$, explaining your deduction from the relationship between the ruler's height and its shadow.
2. b) Apply Thales's Theorem to calculate the actual height of the lamppost with precision.
3. c) Compare the obtained value with your previous estimate and explain why the result is consistent.

Two secant lines, $r$ and $s$, are intersected by three parallel lines ($a$, $b$, and $c$). On line $r$, these parallels determine two consecutive segments: $AB = 3\text{ cm}$ and $BC = 5\text{ cm}$. On line $s$, the corresponding homologous segments are $A'B' = 4.5\text{ cm}$ and $B'C' = x$.

To calculate the value of $x$, Seán wrote the following equality in his notebook:

and after solving for $x$, he obtained $x = 2.7\text{ cm}$.

1. a) Point out what Seán's reasoning error was when matching the segments in the proportion.
2. b) Without doing numerical calculations yet, argue why a result of $2.7\text{ cm}$ contradicts the geometric logic of Thales's Theorem by looking at the measures of $AB$ and $BC$.
3. c) Write the correct proportion, solve it step by step, and determine the actual measure 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 $x$.

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.

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?

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.

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.

A stepladder $2.4$ metres long leans against a wall. The rungs are parallel to the ground. The first rung is $30$ cm from the ground along the slope 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 $1.2$ metres, what is the horizontal distance from the base to the vertical line of the second rung?

To improve thermal comfort at a health center, the City Council commissions the installation of an inclined pergola with solar panels. The inclined main beam is $6\text{ m}$ long and rests from the ground up to the top of a $2.4\text{ m}$ high vertical wall. To reinforce the metallic structure, two auxiliary vertical pillars are installed (parallel to the wall):

• The first pillar is placed $1.5\text{ m}$ from the support on the ground (measured along the inclined beam).

• The second pillar is placed $4\text{ m}$ from that support (also measured along the beam).

1. a) Model the situation by identifying the triangles in Thales's position and justify why the two pillars and the wall meet the conditions of the theorem.
2. b) Calculate the exact height that each of the two vertical reinforcing pillars must have.
3. c) If municipal accessibility regulations require that the minimum clearance height under any passable point of the structure be at least $1.5\text{ m}$, determine with reasons whether an adult walking upright can pass underneath the first pillar.

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.

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 $x + 2$ metres, and the corresponding segments on the second tie rod measure $3$ and $4$ metres, calculate the value of $x$ and the total length of the first tie rod.

A street artist wants to project the shadow of a vertical sculpture $2.5$ metres tall onto a wall located at a certain distance. To do so, they place a spotlight on the ground $6$ metres from the sculpture. If the shadow projected on the wall is exactly $4$ metres tall, how far is the wall from the spotlight? If the artist moves the spotlight $2$ metres 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 $3$ metres 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 $12$ metres and the distance from the bar to the cable's ground anchor is $8$ metres, calculate the height of the small tower. If the large tower is $30$ metres from the ground anchor, what is its height?

To measure the depth of a circular well with a diameter of $2$ metres, a person $1.70$ metres tall stands $1.2$ metres from the edge of the well. From that position, their eyes (located $1.60$ metres 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.

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

A carpenter is building a designer shelving unit in the shape of a right-angled triangle. He has placed two horizontal shelves parallel to the base. The base of the shelving unit measures $1.2$ metres. 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 centimetres).

Two buildings on the same street cast overlapping shadows at a given time of day. Building A is $18$ metres tall and casts a shadow of $12$ metres. 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 $12$ metres, and the distance between the two buildings is $10$ metres, 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-angled trapezium has a longer base of $8$ cm and a shorter base of $5$ cm. If the actual height of this land is $100$ metres, calculate:

1. The actual length of the bases in metres.

2. The height on the map in centimetres.

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.

To measure the height of their town's bell tower during a field trip, two groups of 2nd Year students propose two procedures based on similarity:

• Method 1 (Sun shadow): They stick a $1.5\text{ m}$ vertical rod into the ground, measure its shadow ($0.9\text{ m}$) and the shadow cast by the tower ($15\text{ m}$) at $11:30$ in the morning.

• Method 2 (Mirror on the ground): They place a horizontal plane mirror on the ground $20\text{ m}$ from the base of the tower. A student whose eye height is $1.60\text{ m}$ steps back $1.28\text{ m}$ from the mirror in a straight line until she sees the top of the tower reflected in the center of the mirror (knowing that the angle of incidence of light on the mirror is equal to the angle of reflection).

1. a) Carry out the calculations that each group would perform and check whether both methods yield a matching result for the height of the bell tower.
2. b) Explain the geometric foundation of the mirror method by proving why the two triangles involved are similar and how it relates to Thales's position.
3. c) If the activity had to be carried out on a very overcast day without defined shadows, or in an enclosed square where surrounding buildings block the tower's shadow, which of the two methods would you choose? Justify the advantages, limitations, and drawbacks of each in real-life situations.

On a written test, the following problem is presented: “In a triangle $ABC$, side $AB$ measures $15\text{ cm}$. An interior segment $DE$ is drawn parallel to side $BC$ (with $D$ on $AB$ and $E$ on $AC$), such that $AD = 6\text{ cm}$ and base $BC = 20\text{ cm}$. Calculate the length of parallel segment $DE$”.

Two students discuss the correct way to solve it:

• Aoife claims: “By Thales's Theorem, the segments are proportional. Since the lower segment is $DB = 15 - 6 = 9\text{ cm}$, the proportion is $\frac{AD}{DB} = \frac{DE}{BC}$, that is, $\frac{6}{9} = \frac{DE}{20}$, which gives $DE \approx 13.33\text{ cm}$”.

• Seán counters: “Your formula is wrong because the parallel sides belong to complete triangles, not to segments cut along a transversal. The correct proportion is $\frac{AD}{AB} = \frac{DE}{BC}$, that is, $\frac{6}{15} = \frac{DE}{20}$, which gives $DE = 8\text{ cm}$”.

1. a) Rigorously justify who is right and precisely explain the conceptual error of the mistaken person in confusing the segments on the transversals with the parallel bases.
2. b) To defend Aoife, a third student says: “Aoife's formula does work if $D$ is placed right at the midpoint of $AB$”. Check what would happen with Aoife's formula if $AD = 7.5\text{ cm}$ and $DB = 7.5\text{ cm}$, calculate the value it predicts, and argue why that result geometrically proves that the formula is still nonsense.
3. c) If the similarity ratio is defined as $r = \frac{AD}{AB}$, write the correct general relation that allows calculating the length of segment $DE$ from base $BC$ and $r$.