Exercises: Thales's theorem

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

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

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

Explain the geometric steps needed to divide a $11$ in segment into $5$ exactly equal parts using a ruler, a compass and Thales's theorem.

At a certain time of day, a traffic light casts a shadow and, at the same time, a nearby bollard casts its own. Knowing that the sun's rays strike parallel to each other, calculate the height of the traffic light $H$.

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

In a triangle $\Delta ABC$, a segment $MN$ is drawn parallel to side $BC$, such that $M$ belongs to side $AB$ and $N$ to side $AC$. We know that $AM = 4$ in, $MB = 2$ in and $AN = 6$ in. Calculate the length of the full side $AC$.

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

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

To measure the width of a river with parallel banks, a surveyor places three stakes $A, B$ and $C$ on one of the banks, and takes a tree $T$ on the opposite bank as a reference. Knowing that segment $AB$ is parallel to segment $CD$, that $AB = 8$ ft, $BC = 12$ ft and the perpendicular distance from $A$ to the river is $2$ ft, calculate the width of the river by applying the similarity of triangles.

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

In a tennis match, a player hits the ball at a height of $2.4$ ft. The ball travels in a straight line, exactly grazes the top tape of the net (which is at a height of $0.9$ ft) and lands in the opponent's court. If the horizontal distance from the player to the net is $12$ ft, at what distance from the net will the ball bounce?

Find the value of $x$ by applying Thales's theorem in the following diagram, where the three vertical lines are parallel. Express the result as an integer or a simplified fraction.

During a solar eclipse, the Moon comes exactly between the Earth and the Sun, blocking its light. Knowing that the diameter of the Sun is approximately $1.4 \times 10^6$ miles and it is $1.5 \times 10^8$ miles from the Earth, and that the Moon is $384000$ miles from the Earth, calculate the approximate diameter of the Moon using the similarity of triangles.

Emily, who is $1.65$ ft tall, walks away from a lit street lamp in a straight line. At a given moment, her shadow measures $2.2$ ft. If the distance from the base of the street lamp to the tip of Emily's shadow is $8$ ft, what is the height of the street lamp?

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

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

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

In a trapezoid $ABCD$ with bases $AB = 25$ in and $CD = 15$ in, a segment $MN$ is drawn parallel to the bases, with $M$ on $AD$ and $N$ on $BC$. We know that 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 oblique sides from a vertex).

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

In the structure of a tower crane, the main jib forms a triangle with the tension cables. From a node $A$ on the main jib, a vertical tie is drawn to the lower horizontal jib, measuring $3$ ft. From a node $B$, located $4$ ft further along the main jib, another vertical tie is drawn measuring $4.5$ ft. At what distance from node $A$ is the articulation point (where the two jibs of the crane meet)?

Two parallel lines are at a distance of $H$ from each other. Two transversal segments intersect at a point $P$ between the parallel lines, forming an hourglass shape. The segments of one of the transversals measure $x$ and $x+5$, while the base of the upper triangle measures $12$ in and the base of the lower triangle measures $27$ in. 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 $2$ ft high. They drive the first rod into the ground and, lying down, visually align the tip of the rod with the top of the mountain. The distance from their eye to the rod is $1.5$ ft. Then they move back $150$ ft in a straight line, drive the second rod in, and again align their eye with the tip of the rod and the top. This time, the distance from their eye to the rod is $1.6$ ft. Calculate the height of the mountain.

Prove, using the converse of Thales's theorem, that the 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 $L$? Argue your answer based on the similarity of triangles.