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 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 center to the eaves is $5$ m). The maximum height at the center is $3$ m. If the strut is placed $1.5$ m 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 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 theater 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 plane mirror is placed on the horizontal ground. A person whose eyes are $1.60$ m above the ground stands $1.20$ m from the center of the mirror and sees the antenna on top 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 antenna 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.