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 can't climb it with a tape measure. But on a sunny day, all you need is a friend, a tape measure and one idea: parallel lines chop other lines into proportional pieces. In this section you will learn that idea, its converse, and how surveyors, photographers and astronomers use it to measure things they can't reach.
Parallel lines and proportional segments
A transversal is a line that crosses other lines. When parallel lines cut two transversals, the pieces on one transversal are in the same ratios as the pieces on the other.
Some problems call this Thales's theorem. In this course we call it the Three Parallel Lines Theorem.
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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 $s$?
A triangle is the simplest place to see parallel lines at work. A segment parallel to one side "splits" the other two sides into proportional pieces. That's why it's also called the side-splitter theorem.
Slide the scale factor below to see how a smaller triangle stays similar to a bigger one.
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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 $AC$.
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.
The theorem goes one way: parallel $\Rightarrow$ proportional. The converse goes the other way, and it lets you prove lines are parallel just from measurements. No protractor needed.
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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 Three Parallel Lines Theorem also gives a construction. With a straightedge and compass you can split any segment into any number of equal parts, with no decimals and no measuring.
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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.
The sun is so far away that its rays arrive essentially parallel. So at the same moment, every vertical object and its shadow form a right triangle with the same angles. These triangles are similar by AA, and their sides are proportional.
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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?
Light through a point: pinhole cameras, eclipses and spotlights
Light travels in straight lines. If the rays all pass through one point (a tiny hole, your eye, a spotlight), the object and its image form two similar triangles that share that point as a vertex.
The same idea explains a total solar eclipse. The Moon exactly covers the Sun, so the Moon's diameter is to its distance from you as the Sun's diameter is to its distance from you.
A spotlight on the floor works the same way. The actor and his shadow on the backdrop are two similar right triangles with the spotlight at the shared vertex. Each triangle's height goes with its distance from the light.
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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 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 ($h_1 - h_2$).
Surveyors measure things they can't reach by building a small, measurable triangle that is similar to a big one. Here are three classic setups.
Mirror. A ray of light bounces off a flat mirror at the same angle it arrived. So the right triangle from your eyes to the mirror is similar by AA to the right triangle from the mirror to the top of what you see.
River. To find a river's width, build a pair of similar triangles with the width as one side.
Mountain, two sightings. When you can't reach the base of a tall object, you don't know its distance. So use two sightings and let that distance be a variable.
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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.
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?
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.
Any straight slanted line crossing regular vertical supports (a ramp, a roof, the path of a ball) creates similar right triangles. The trick is to put the right triangle's vertex where the slanted line meets the ground, and measure every horizontal distance from there.
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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?
Sometimes the unknown shows up in two places of the same proportion, such as a segment $x$ and the whole $x + 6$. Then you get an equation to solve. It can be linear or quadratic.
The hourglass. When two transversals cross between two parallel lines, the two triangles are similar. The vertical angles at the crossing point are congruent and the alternate interior angles are congruent, so AA works.
Apex setups. Two parallel braces hang from a slanted beam. If their lengths are $p$ and $q$, the distances from the apex (where beam and floor meet) are in the same ratio $p : q$. If the braces are $6$ ft apart along the beam, with $p = 2$ and $q = 5$, call the distance from the apex to the shorter brace $d$. Then $\dfrac{d}{d+6} = \dfrac{2}{5}$, so $5d = 2d + 12$ and $d = 4$ ft.
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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.
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)?
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.
A midsegment of a triangle connects the midpoints of two sides. It is the special case of the Triangle Proportionality Theorem where the ratio is $\dfrac{1}{2}$.
Trapezoids. A segment parallel to the bases of a trapezoid isn't in a triangle, so build one. Draw a line through a vertex parallel to a leg. It splits the trapezoid into a parallelogram and a triangle.
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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.
Back to the flagpole: it's the shadow example. Emma's shadow, the pole's shadow and the sun's parallel rays give two similar triangles, so a tape measure on the ground is all you need.
Summary
Let's practice! 8 medium exercises in this section