Thales' Theorem
Parallel lines cutting two intersecting lines: it looks like an abstract geometry drawing, but it is the tool that surveyors, photographers, and even astronomers use to measure without getting any closer. In this section you will get to know Thales' theorem in depth: what it says, its converse, and its most practical applications.
Thales' theorem, stated precisely
Find the value of the unknown segment $x$ in the following figure, given that the horizontal lines are parallel to each other.
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…
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.
Triangles in Thales position
When the parallel segment cuts the two sides of a triangle, the segments on each side are proportional. It is the same theorem, applied to a triangle instead of two loose lines.
The triangles in the figure are in Thales' position. Calculate the lengths of sides $x$ and $y$.
The triangles in the figure are in Thales' position. Calculate the lengths of sides $x$ and $y$.
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…
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.
The converse of Thales' theorem
This lets you check whether lines are parallel from measurements alone, without needing a protractor or a set square.
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…
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 $s$?
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,…
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.
Dividing a segment into equal parts
Thales' theorem does not just calculate lengths: it is the straightedge-and-compass method for dividing any segment into exactly equal parts, without having to measure to decimal precision.
Explain the geometric steps required to divide a $11$ cm segment into $5$ exactly equal parts using a straightedge, compass and Thales' Theorem.
Explain the geometric steps required to divide a $11$ cm segment into $5$ exactly equal parts using a straightedge, compass and Thales' Theorem.
Shadows, pinhole cameras, and mirrors
A ray of light passing through a fixed point (a pinhole, the center of a mirror) creates similar triangles between the real object and its image or shadow. This phenomenon is the basis of the pinhole camera, photography, and measuring with mirrors.
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,…
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.
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 $1.4 \times 10^6$…
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.
Ramps, nets, and structures
Any element that runs in a straight line and crosses regularly spaced vertical supports (a ramp, a tennis net, a roof) forms triangles in Thales position with the intermediate supports.
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…
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…
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?
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…
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?
The midsegment theorem
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…
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.
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…
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.