Exercises: Thales' Theorem

Find the value of the unknown segment $x$ in the following figure, given that the horizontal lines are parallel to each other.

Two intersecting lines are cut by two parallel lines. On the first line, the segments determined by the parallels measure $12$ cm and $15$ cm. On the second line, the segment corresponding to the $12$ cm segment measures $8$ cm. Calculate the length of the segment corresponding to the $15$ cm segment.

The triangles in the figure are in Thales' position. Calculate the lengths of sides $x$ and $y$.

Explain the geometric steps required to divide a $11$ cm segment into $5$ exactly equal parts using a straightedge, compass and Thales' Theorem.

Three parallel lines are cut by two intersecting lines $r$ and $s$. On line $r$, the segments measure $3.5$ cm and $5$ cm. On line $s$, the segment corresponding to the $3.5$ cm segment measures $4.2$ cm. How long is the segment corresponding to the $5$ cm segment on line $s$?

In a triangle $\Delta ABC$, a line segment $MN$ is drawn parallel to side $BC$, such that $M$ lies on side $AB$ and $N$ on side $AC$. Given that $AM = 4$ cm, $MB = 2$ cm and $AN = 6$ cm, calculate the length of the complete side $AC$.

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

Check whether the segments determined by three lines on two intersecting lines are proportional. On the first line, they measure $15$ cm and $20$ cm. On the second line, they measure $24$ cm and $32$ cm. Are the three lines parallel? Justify your answer using the converse of Thales' Theorem.

In a triangle $ABC$, a segment $DE$ is drawn parallel to side $BC$, with point $D$ located on side $AB$ and point $E$ on side $AC$. The following measurements are known: $AD = 6\text{ cm}$, $DB = 3\text{ cm}$, and the base $BC = 12\text{ cm}$. Darragh tries to calculate the length of the parallel segment $DE$ and sets up the following in his notebook:

1. a) Without performing any numerical calculation, explain what immediate geometric evidence shows that the result of $24\text{ cm}$ obtained by Darragh is impossible.
2. b) Identify the conceptual error that Darragh made when setting up the proportion between the lateral sides and the parallel bases.
3. c) Write the correct proportion applying the similarity relationships derived from Thales's theorem and find the actual length of segment $DE$.

In an urban park, a triangular flowerbed $ABC$ is designed. On side $AB$, which measures $15\text{ m}$, a sprinkler is placed at point $M$, at a distance of $9\text{ m}$ from vertex $A$. A secondary pipe $MN$ is laid parallel to side $BC$, so that point $N$ lies on the other side, $AC$, whose total length is $14\text{ m}$.

1. a) Before performing any exact calculation, estimate whether segment $AN$ will measure more or less than half of side $AC$ ($7\text{ m}$) and reason between which two integer values it should approximately lie, observing what proportion $AM$ represents relative to the total length of $AB$.
2. b) Apply Thales's theorem to calculate the exact length of segment $AN$ and that of the remaining segment $NC$.
3. c) Check whether the value obtained matches your initial estimate and explain why the ratio $\dfrac{AM}{MB}$ must numerically match the ratio $\dfrac{AN}{NC}$.

In a pinhole camera, light enters through a small hole and projects an inverted image of an object onto the back wall. If a tree $4$ m tall is $15$ m from the hole, calculate the height of the projected image $h$ if the depth of the camera is $0.3$ m.

To measure the width of a river with parallel banks, a surveyor places three stakes $A, B$, and $C$ aligned in that order on one of the banks, positioning $A$ directly opposite a tree $T$ on the other bank (so that the segment $TA$ is perpendicular to the banks), and a fourth stake $D$ such that the points $T, B$, and $D$ are aligned. Knowing that the segment $TA$ is parallel to the segment $CD$, that $AB = 8$ m, $BC = 12$ m, and the perpendicular distance from $D$ to the river is $2$ m, calculate the width of the river using similar triangles.

A wheelchair access ramp overcomes a vertical drop of $1.2$ m over a horizontal distance of $15$ m. If an intermediate vertical support is to be placed at a horizontal distance of $6$ m from the start of the ramp, what must be the height of this support?

In a tennis match, a player hits the ball at a height of $2.4$ m. The ball travels in a straight line, just brushes the top of the net (which is $0.9$ m high), and lands on the opponent's court. If the horizontal distance from the player to the net is $12$ m, how far from the net will the ball bounce?

Find the value of $x$ using Thales' Theorem in the following diagram, where the three vertical lines are parallel. Express the answer as an integer or a simplified fraction.

During a solar eclipse, the Moon passes directly between the Earth and the Sun, blocking its light. Given that the diameter of the Sun is approximately $1.4 \times 10^6$ km and it is $1.5 \times 10^8$ km from the Earth, and that the Moon is $384000$ km from the Earth, calculate the approximate diameter of the Moon using similar triangles.

In the structure of a gabled roof, a vertical reinforcing strut is placed. The base of the roof measures $10$ m in total (from the centre to the eaves is $5$ m). The maximum height at the centre is $3$ m. If the strut is placed $1.5$ m from the centre towards the eaves, what must its length be?

On the urban planning map of a municipality, two straight avenues branch off from the same roundabout $O$. To connect both avenues, the town hall plans three straight, mutually parallel pedestrian crossings: Crossing 1, Crossing 2, and Crossing 3.

Along the first avenue, the distance from roundabout $O$ to Crossing 1 is $40\text{ m}$; the distance between Crossing 1 and Crossing 2 is $24\text{ m}$; and the distance between Crossing 2 and Crossing 3 is $32\text{ m}$.

Along the second avenue, a surveying team measures that the total distance from roundabout $O$ to Crossing 3 is exactly $120\text{ m}$.

1. a) Model the situation by identifying which geometric elements correspond to the secant lines and which to the parallel lines of Thales's theorem.
2. b) Calculate the length of each of the three sections into which the second avenue is divided by the pedestrian crossings (from the roundabout to Crossing 1, from Crossing 1 to Crossing 2, and from Crossing 2 to Crossing 3).
3. c) Municipal accessibility regulations recommend that the distance between two consecutive pedestrian crossings should not exceed $35\text{ m}$. Determine with justification whether the section between Crossing 2 and Crossing 3 along the second avenue complies with this recommendation.

The metallic structure of a triangular glass window bounded by two secant profiles $r$ and $s$, joined by three horizontal parallel cross-braces $a$, $b$, and $c$, is to be designed. These cross-braces define two adjacent sections $AB$ and $BC$ along profile $r$, and two adjacent sections $A'B'$ and $B'C'$ along profile $s$, satisfying Thales's theorem with a pre-established design ratio of $\dfrac{AB}{BC} = \dfrac{A'B'}{B'C'} = \dfrac{2}{5}$.

1. a) If the total length of the glazed section along the first profile must be $AC = 35\text{ cm}$, deduce the exact values that segments $AB$ and $BC$ must have.
2. b) For the second profile $s$, propose two different integer lengths for $A'B'$ and $B'C'$ such that their total sum is strictly greater than $60\text{ cm}$ and they preserve the required proportion exactly.
3. c) Write a general mathematical expression that allows you to directly calculate the length of the upper segment $A'B'$ in terms of the total length $L = A'C'$ of profile $s$, valid for any structure that must maintain this same design proportion.

In a system of two intersecting lines cut by three parallel lines, the segments on the first line measure $x$ and $x+3$. On the second line, the corresponding segments measure $x-2$ and $x$. Set up the proportion, solve the resulting equation, and find the values of all four segments.

A theatre spotlight shines on an actor, projecting his shadow onto the backdrop. The spotlight is at ground level. The actor, who is $1.80$ m tall, walks towards the backdrop. When he is $4$ m from the spotlight, his shadow measures $h_1$. If he walks $2$ m further towards the backdrop, his shadow measures $h_2$. Knowing that the total distance from the spotlight to the backdrop is $10$ m, calculate the difference in height between the two projected shadows ($h_1 - h_2$).

In a trapezium $ABCD$ with parallel sides (bases) measuring $AB = 25$ cm and $CD = 15$ cm, a segment $MN$ is drawn parallel to the bases, with $M$ on $AD$ and $N$ on $BC$. It is known that the segment $MN$ divides the non-parallel sides in the ratio $\frac{AM}{MD} = \frac{2}{3}$. Calculate the length of segment $MN$. (Hint: Draw a line parallel to one of the non-parallel sides from a vertex).

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

A flat mirror is placed on the horizontal ground. A person whose eyes are $1.60$ m above the ground stands $1.20$ m from the centre of the mirror and sees the aerial of a building reflected exactly in it. If the distance from the mirror to the base of the building is $18$ m, and both the person and the building are perpendicular to the ground, what is the total height of the aerial from the ground?

In the structure of a tower crane, the jib forms a triangle with the tension cables. From a node $A$ on the jib, a vertical tie is drawn to the lower horizontal arm, measuring $3$ m. From a node $B$, located $4$ m further along the jib, another vertical tie is drawn measuring $4.5$ m. How far from node $A$ is the pivot point (where the two arms of the crane meet)?

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

Two parallel lines are a distance $H$ apart. Two transversal line segments cross at a point $P$ between the parallels, forming an hourglass shape. The segments of one of the transversals measure $x$ and $x+5$, while the base of the top triangle measures $12$ cm and the base of the bottom triangle measures $27$ cm. Calculate the value of $x$ and the total length of that transversal.

To measure the height of an inaccessible mountain without using trigonometry, a surveyor uses two vertical rods $2$ m tall. He drives the first rod into the ground and, lying down, visually aligns the top of the rod with the peak of the mountain. The distance from his eye to the rod is $1.5$ m. He then moves back $150$ m in a straight line, drives in the second rod, and aligns his eye with the top of the rod and the peak once again. This time, the distance from his eye to the rod is $1.6$ m. Calculate the height of the mountain.

Prove, using the converse of Thales' Theorem, that the line segment joining the midpoints of two sides of any triangle is parallel to the third side. If the third side measures $L$, what is the length of this segment in terms of $L$? Justify your answer based on similar triangles.

In a wooden triangular structure $ABC$, the lateral sides measure $AB = 15\text{ cm}$ and $AC = 20\text{ cm}$. To reinforce it, an internal slat $MN$ is placed connecting a point $M$ on side $AB$ with a point $N$ on side $AC$, such that $AM = 6\text{ cm}$ (leaving $MB = 9\text{ cm}$) and $AN = 8\text{ cm}$ (leaving $NC = 12\text{ cm}$). We want to verify whether slat $MN$ has ended up strictly parallel to the base $BC$.

• Method A: Check the equality of ratios between adjacent segments: $\dfrac{AM}{MB} \stackrel{?}{=} \dfrac{AN}{NC}$.

• Method B: Check the equality of ratios between the partial segment and the full side: $\dfrac{AM}{AB} \stackrel{?}{=} \dfrac{AN}{AC}$.

1. a) Apply both methods numerically and determine whether slat $MN$ is parallel to the base $BC$ according to the converse of Thales's theorem.
2. b) Prove algebraically that, if in any triangle the equality of Method A holds (setting $AM = x$, $MB = a - x$, $AN = y$, $NC = b - y$), the equality of Method B necessarily holds, justifying why both criteria are mathematically equivalent.
3. c) If the base measures $BC = 25\text{ cm}$ and you wish to find the length of slat $MN$, justify which of the two methods links directly to triangle similarity to calculate $MN$ and explain what conceptual error someone would make if they tried to calculate it from the proportion in Method A.

A triangular rural plot $ABC$ borders a straight irrigation canal along side $AC$ ($180\text{ m}$) and a straight path along side $AB$ ($150\text{ m}$). The third boundary, $BC$, measures $210\text{ m}$. In a reorganization project, technicians lay out two straight dividing fences, $D_1E_1$ and $D_2E_2$, both parallel to boundary $BC$, placing points $D_1$ and $D_2$ along path $AB$ so that it is divided into three equal parts of $50\text{ m}$ each ($D_1$ closest to vertex $A$).

1. a) Model the position of anchors $E_1$ and $E_2$ along canal $AC$ by calculating the length of the three resulting sections on that boundary using Thales's theorem.
2. b) Calculate the exact length of each of the two interior dividing fences ($D_1E_1$ and $D_2E_2$), explaining in detail why the ratio of segments along the sides $\dfrac{AD_1}{D_1D_2}$ cannot be used directly, and which geometric ratio must be used instead.
3. c) One of the owners states: "Since the fences divide both lateral boundaries into three parts of equal length, the entire plot has been divided into three areas of equal surface." Determine with justification whether this statement is true or false by comparing the area of the upper triangle $AD_1E_1$ with the area of the lower trapezoid bounded by $D_2E_2$ and boundary $BC$ using the similarity ratio.