Before you start, make sure you know: Ratio (Math 7)
Before you start, make sure you know: Similarity (Math 8 (Pre-Algebra))
How tall is the flagpole in front of your school? You have no ladder and no tape measure, just a sunny afternoon and a yardstick. Parallel lines that cut across other lines create proportional segments, and that fact lets surveyors, photographers, and even astronomers measure things without ever touching them. By the end of this page you'll be able to answer the flagpole question.
Parallel Lines Cut Transversals Proportionally
A transversal is a line that crosses other lines. When parallel lines cut two transversals, they chop both transversals into pieces, and those pieces are in the same ratio.
Some problems call this Thales's theorem. In US classes it appears as a corollary of the Triangle Proportionality Theorem, and that is how we'll refer to it here.
In the figure, $\dfrac{6}{4.5} = \dfrac{8}{x}$, so $6x = 36$ and $x = 6$.
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Calculate the value of the unknown segment $x$ in the following figure, knowing that the horizontal lines are parallel to each other.
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…
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.
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…
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 $s$?
Now draw one line parallel to a side of a triangle. It cuts the other two sides, and those sides are split proportionally. It is the same idea as the last section, with the two transversals meeting at a vertex. Problems often describe this setup as a triangle cut by a line parallel to one side.
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The triangles in the figure are in Thales's position. Calculate the lengths of the sides $x$ and $y$.
The triangles in the figure are in Thales's position. Calculate the lengths of the sides $x$ and $y$.
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…
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 $AC$.
Often the unknown appears inside the lengths themselves, such as $x$ and $x + 5$. The plan is the same: match the segments, write the proportion, cross-multiply, and solve. Two things to watch for:
If the $x^2$ terms cancel, you get a linear equation. Otherwise you get a quadratic: move everything to one side and factor.
Lengths are positive, so throw out any solution that makes a segment zero or negative.
Two transversals that cross between two parallel lines make an hourglass. The two triangles are similar by AA, because the vertical angles at the crossing point are equal and the alternate interior angles are equal.
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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…
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.
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,…
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.
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.…
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.
The theorem also works backward. That lets you prove lines are parallel from measurements alone, with no protractor.
For lines cut by two transversals, the test is the same. Compare corresponding ratios, and if they all match, the lines are parallel.
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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…
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.
The theorem builds more than measurements: it gives a compass-and-straightedge construction that splits any segment into equal parts, with no decimals.
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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.
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.
Sunlight reaches Earth in (essentially) parallel rays. An object and its shadow form a right triangle, and every object at the same moment makes a right triangle with the same angle of the sun. By AA, those triangles are similar.
Streetlights and stage spotlights are different: the light comes from one point, and the rays spread out. The similar triangles now share that point as a vertex.
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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…
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 $H$.
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…
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?
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…
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 ($h_1 - h_2$).
Light travels in straight lines. When rays pass through a tiny hole, bounce off a mirror, or skim past a coin, they build similar triangles between the real object and the image or shadow.
A mirror works because of the law of reflection: the angle of incidence equals the angle of reflection. With the two right angles at the ground, the two triangles are similar (AA).
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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…
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.
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 $1.4 \times 10^6$…
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.
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…
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?
Anything that rises in a straight line past evenly placed vertical supports (a ramp, a roof, the path of a ball) creates similar right triangles that start at the same vertex.
For a roof, take the peak as the vertex. A vertical strut at a given horizontal distance from the center is part of a smaller triangle with the same peak, so measure from the peak (or from the center line), not from the eaves.
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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…
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…
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?
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…
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?
Surveyors measure rivers, cliffs, and mountains without crossing or climbing them. They build triangles they can measure that are similar to the ones they can't.
A river. Put stakes on your bank and pick a landmark $T$ on the far bank. Lines from $T$ to the stakes make triangles with a segment you can measure parallel to the bank. Find the pair of similar triangles that share the vertex $T$.
A tower or crane. Vertical supports that stand on a horizontal line below a sloping beam have heights proportional to their distances from the vertex where the beam meets the horizontal.
What if the object is so far away that you don't know either the distance or the height? Then take two sightings. Lie on the ground so your eye is at ground level, and line up the top of a vertical rod with the top of the mountain.
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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…
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.
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…
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)?
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…
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.
What if the line cuts the sides exactly in half? The special case is useful enough to have its own name.
A trapezoid problem is solved by making a triangle: draw a line through one vertex parallel to a leg.
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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…
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).
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…
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.