Similar Right Triangles
Emma is 5 ft tall and her shadow on the soccer field is 4 ft long. At the same moment the school flagpole casts a shadow 24 ft long. Nobody is climbing that pole with a tape measure, so how tall is it? The answer comes from similar right triangles, and the same idea also gives two handy formulas for any right triangle that has its altitude drawn to the hypotenuse.
Similar Right Triangles and Scale Factor
Two triangles are similar when their angles match. For right triangles this is easy to check: both already have a $90^\circ$ angle, so if they share one acute angle, they are similar by AA.
When two triangles are similar, every pair of corresponding sides has the same ratio. That ratio is the scale factor $k$.
Two right triangles are similar. The first has legs measuring $3$ in and $4$ in. If the shorter leg of the second triangle measures $9$ in, how long is the hypotenuse of…
Two right triangles are similar. The first has legs measuring $3$ in and $4$ in. If the shorter leg of the second triangle measures $9$ in, how long is the hypotenuse of this second triangle?
Two set squares (which have the shape of a scalene right triangle) are similar. If the hypotenuse of the larger set square is three times that of the smaller set square,…
Two set squares (which have the shape of a scalene right triangle) are similar. If the hypotenuse of the larger set square is three times that of the smaller set square, how many times larger is the area of the large set square compared to the small one?
Shadows, Lamps, and Mirrors
You can measure things you cannot reach by finding two similar right triangles. The sun's rays are (almost exactly) parallel, so a person and a flagpole make the same angle with the sun. Each stands straight up from the ground, so each makes a right angle. That is AA, and the triangles are similar.
A streetlamp works the same way, but be careful: the big triangle includes the distance to the lamp and the shadow.
A flat mirror on the ground also creates similar triangles. Light reflects at the same angle it arrives, so the angle at the mirror is the same on both sides. Add the right angles at the ground and you have AA.
A vertical post $2$ ft high casts a shadow of $1,5$ ft on the ground. At the same time and under the same conditions, how long will the shadow cast by a…
A vertical post $2$ ft high casts a shadow of $1,5$ ft on the ground. At the same time and under the same conditions, how long will the shadow cast by a telecommunications tower $40$ ft high be?
A street lamp $4,5$ ft high illuminates a person $1,8$ ft tall who is standing $3$ ft from the base of the lamp. Use the similarity of right triangles to find out the…
A street lamp $4,5$ ft high illuminates a person $1,8$ ft tall who is standing $3$ ft from the base of the lamp. Use the similarity of right triangles to find out the length of the shadow the person casts on the ground.
To measure the height of a building, a student places a flat mirror on the ground $15$ ft from the base of the building. Stepping back $1,2$ ft from the mirror, they see…
To measure the height of a building, a student places a flat mirror on the ground $15$ ft from the base of the building. Stepping back $1,2$ ft from the mirror, they see the top of the building reflected in it. If the student's eyes are $1,6$ ft from the ground, what is the height of the building?
Ramps and Pinhole Cameras
The same move works in many places: find the right triangles that share an angle, match corresponding legs, and write a proportion.
| Situation | The similar triangles | Proportion |
|---|---|---|
| Ramp | A small triangle under the ramp and the whole ramp triangle | $\dfrac{\text{height so far}}{\text{run so far}} = \dfrac{\text{total rise}}{\text{total run}}$ |
| Pinhole camera | Object and image triangles meeting at the pinhole | $\dfrac{\text{object height}}{\text{distance to object}} = \dfrac{\text{image height}}{\text{depth of camera}}$ |
| Pool table | Two triangles on either side of the cushion bounce | $\dfrac{\text{sideways distance 1}}{\text{sideways distance 2}} = \dfrac{\text{lengthwise 1}}{\text{lengthwise 2}}$ |
| Cone (glass) | Cross-section through the axis | $\dfrac{\text{water radius}}{\text{rim radius}} = \dfrac{\text{water height}}{\text{total height}}$ |
A pinhole camera is a dark box with a tiny hole in the front. Light from the top of an object passes straight through the hole and lands at the bottom of the back wall, so the image is upside down. The object and the image form two right triangles that meet at the pinhole, and they are similar because their vertical angles at the hole are equal.
Ramps work the same way: the small triangle under the ramp and the whole ramp triangle share the angle at the bottom.
An accessibility ramp has a total length of $12$ ft resting on the ground and overcomes a maximum height difference of $1,5$ ft. If a wheelchair has advanced $4$ ft up…
An accessibility ramp has a total length of $12$ ft resting on the ground and overcomes a maximum height difference of $1,5$ ft. If a wheelchair has advanced $4$ ft up the ramp from its start at the bottom, at what height is it relative to ground level?
In a homemade pinhole camera $20$ in deep, an inverted image of a tree located $15$ ft from the pinhole is formed. If the image formed at the back of the camera has a…
In a homemade pinhole camera $20$ in deep, an inverted image of a tree located $15$ ft from the pinhole is formed. If the image formed at the back of the camera has a height of $8$ in, what is the real height of the tree in feet?
Pool Tables and Cones
For a cone, cut it in half along its axis. The water's surface is parallel to the rim, so the small triangle (axis, water radius) and the big triangle (axis, rim radius) share the angle at the bottom tip.
On a rectangular billiard table, a ball is hit without spin against the opposite cushion, forming two similar right triangles in its path. The ball starts at $40$ in…
On a rectangular billiard table, a ball is hit without spin against the opposite cushion, forming two similar right triangles in its path. The ball starts at $40$ in from the side cushion and rebounds at $60$ in from the same cushion. If the total longitudinal distance the ball has advanced along the table is $150$ in, determine the longitudinal distances traveled before and after the rebound.
A conical glass has a total depth of $15$ in and a radius of $6$ in at its top opening. It is filled with water until the liquid reaches a height of $10$ in measured…
A conical glass has a total depth of $15$ in and a radius of $6$ in at its top opening. It is filled with water until the liquid reaches a height of $10$ in measured from the bottom (lower vertex). Calculate the radius of the circular surface of the water using the similarity of right triangles formed with the central axis of the glass.
Three Similar Triangles in One
Draw the altitude from the right angle to the hypotenuse. It cuts the right triangle into two smaller right triangles, and all three triangles are similar.
Here is the notation. In right triangle $ABC$ the right angle is at $C$. The hypotenuse is $c = AB$, and the legs are $a = BC$ and $b = AC$. Point $D$ is where the altitude $CD$ meets the hypotenuse, and $h = CD$. The two pieces of the hypotenuse are $m = AD$ and $n = DB$. Each piece is the projection of the neighboring leg onto the hypotenuse, so $m$ goes with $b$ and $n$ goes with $a$.
The hypotenuse is the sum of the two segments: $c = m + n$.
In a right triangle, the projections of the two legs onto the hypotenuse measure $5$ in and $12$ in, respectively. How long is the hypotenuse of the triangle?
In a right triangle, the projections of the two legs onto the hypotenuse measure $5$ in and $12$ in, respectively. How long is the hypotenuse of the triangle?
The Altitude Theorem
Compare the two small triangles, $\triangle ACD \sim \triangle CBD$. Corresponding sides are proportional, so the short leg over the long leg of each is the same ratio: $\dfrac{m}{h} = \dfrac{h}{n}$. Cross-multiplying gives the altitude theorem.
In a right triangle, the altitude drawn to the hypotenuse divides it into two segments measuring $4$ in and $9$ in. Calculate the length of this altitude using the…
In a right triangle, the altitude drawn to the hypotenuse divides it into two segments measuring $4$ in and $9$ in. Calculate the length of this altitude using the corresponding theorem.
The altitude relative to the hypotenuse of a right triangle measures $6$ in. If one of the segments into which it divides the hypotenuse measures $3$ in, how long is the…
The altitude relative to the hypotenuse of a right triangle measures $6$ in. If one of the segments into which it divides the hypotenuse measures $3$ in, how long is the other segment of the hypotenuse?
The Leg Theorem
Now compare a small triangle with the whole triangle. From $\triangle ACD \sim \triangle ABC$ we get $\dfrac{m}{b} = \dfrac{b}{c}$ (the short piece over the leg equals the leg over the hypotenuse), so $b^2 = m \cdot c$. The other leg works the same way.
The hypotenuse of a right triangle measures $25$ in and the projection of one of its legs onto it measures $9$ in. Calculate the length of this leg applying the leg…
The hypotenuse of a right triangle measures $25$ in and the projection of one of its legs onto it measures $9$ in. Calculate the length of this leg applying the leg theorem.
A leg of a right triangle measures $8$ in and its orthogonal projection onto the hypotenuse measures $3,2$ in. Determine the total length of the hypotenuse.
A leg of a right triangle measures $8$ in and its orthogonal projection onto the hypotenuse measures $3,2$ in. Determine the total length of the hypotenuse.
Applications: Trusses, Sails, and Arches
Whenever a right triangle has an altitude drawn to its hypotenuse (a roof truss, a triangular sail, an arch), these theorems give lengths you cannot measure directly.
An angle inscribed in a semicircle is a right angle. So if you connect the two ends of the diameter to any point on the arc, you get a right triangle, and the vertical line from that point down to the diameter is its altitude.
In the design of a truss for a pitched roof in the shape of a right triangle, the vertical beam (which acts as the altitude to the hypotenuse) measures $2,4$ ft and…
In the design of a truss for a pitched roof in the shape of a right triangle, the vertical beam (which acts as the altitude to the hypotenuse) measures $2,4$ ft and separates the lower beam (hypotenuse) into two sections, one of which measures $1,8$ ft. Calculate the total perimeter of the outer structure of the truss.
The main sail of a sailboat has the shape of a right triangle. The seam that marks the altitude drawn from the vertex of the right angle to the hypotenuse measures $6$…
The main sail of a sailboat has the shape of a right triangle. The seam that marks the altitude drawn from the vertex of the right angle to the hypotenuse measures $6$ ft. If the projection of the shorter side of the sail onto the hypotenuse is $4,5$ ft, calculate the total area of the sail in square feet.
A stone bridge arch has the shape of a perfect semicircle with a diameter of $20$ ft. At $4$ ft from one of the ends of the bridge, a vertical pillar needs to be placed…
A stone bridge arch has the shape of a perfect semicircle with a diameter of $20$ ft. At $4$ ft from one of the ends of the bridge, a vertical pillar needs to be placed reaching from the ground to the arch. Using the altitude theorem, determine the exact length this pillar must have.
Setting Up Equations
Sometimes the problem does not give $m$ and $n$ directly. It gives a ratio, an expression with $x$, or the area. Name the unknowns, then use these facts:
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$h^2 = m \cdot n$,
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$c = m + n$,
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area $= \tfrac12 \, c \, h$,
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leg theorem: $a^2 = n \cdot c$ and $b^2 = m \cdot c$.
In a plot of land shaped like a right triangle, an irrigation trench has been dug that coincides with the altitude to the hypotenuse and measures $12$ ft. If the ratio…
In a plot of land shaped like a right triangle, an irrigation trench has been dug that coincides with the altitude to the hypotenuse and measures $12$ ft. If the ratio between the two segments into which this trench divides the hypotenuse is $\frac{9}{16}$, calculate the dimensions of the three sides that bound the land.
In a right triangle, the projection of the shorter leg onto the hypotenuse measures $x$ in, the projection of the longer leg measures $(x+5)$ in, and the altitude…
In a right triangle, the projection of the shorter leg onto the hypotenuse measures $x$ in, the projection of the longer leg measures $(x+5)$ in, and the altitude relative to the hypotenuse measures $(x+2)$ in. Set up the corresponding equation, find the value of $x$, and calculate the total area of the triangle.
Equations with Area and Squared Ratios
When the area is given, use area $= \tfrac12 c h$ to find the hypotenuse first. That gives you $m + n$ and $m \cdot n$, and you can solve for $m$ and $n$ with a substitution.
Here is a nice shortcut. Divide the two leg formulas:
$$\frac{a^2}{b^2} = \frac{n \cdot c}{m \cdot c} = \frac{n}{m}$$The ratio of the squares of the legs equals the ratio of the segments they cut off.
The area of a solar panel shaped like a right triangle is $150$ in$^2$ and the altitude relative to its hypotenuse measures $12$ in. By setting up a system of equations…
The area of a solar panel shaped like a right triangle is $150$ in$^2$ and the altitude relative to its hypotenuse measures $12$ in. By setting up a system of equations with the projections of the legs, determine the exact length of the two legs of the solar panel.
In a right triangle, the ratio between the squares of the legs is $\frac{b^2}{c^2} = \frac{9}{16}$. If the hypotenuse measures $50$ in, use the proportionality…
In a right triangle, the ratio between the squares of the legs is $\frac{b^2}{c^2} = \frac{9}{16}$. If the hypotenuse measures $50$ in, use the proportionality relationship deduced from the leg theorems to first find the measure of the projections of both legs onto the hypotenuse, and subsequently the exact length of the legs.
Challenge Problems with Similar Right Triangles
Three classic problems finish the page. All three come from drawing the right triangles and matching sides.
Two posts and two cables
Two vertical posts of heights $a$ and $b$ stand $d$ apart. A taut cable runs from the base of each post to the top of the other. The cables cross at height $h$.
A square in a right triangle
The distance to the horizon
A line of sight that just touches the Earth is tangent to it, and a tangent line is perpendicular to the radius at the point of tangency. So the center of the Earth $O$, the point on the horizon $T$, and the viewer $B$ form a right triangle with the right angle at $T$. The legs are $R$ (Earth's radius) and $d$ (the distance to the horizon), and the hypotenuse is $R + h$.
Two vertical posts of heights $a = 3$ ft and $b = 6$ ft are separated by an unknown distance. Two taut cables are stretched from the base of each post to the top of the…
Two vertical posts of heights $a = 3$ ft and $b = 6$ ft are separated by an unknown distance. Two taut cables are stretched from the base of each post to the top of the other post. Using the similarity of the right triangles formed, show that the height $h$ at which the cables cross is independent of the distance separating the posts and calculate its value.
A square is inscribed in a right triangle with legs of $8$ in and $6$ in, such that one of its vertices coincides with the right angle of the triangle, and the two sides…
A square is inscribed in a right triangle with legs of $8$ in and $6$ in, such that one of its vertices coincides with the right angle of the triangle, and the two sides of the square forming that vertex rest on the legs. Calculate the measure of the side of the square $x$ using the similarity of the right triangles formed.
If we consider the Earth to be a sphere of radius $R = 6371$ miles, and a person is in a hot air balloon at a height of $h = 2$ miles above the surface, the line of…
If we consider the Earth to be a sphere of radius $R = 6371$ miles, and a person is in a hot air balloon at a height of $h = 2$ miles above the surface, the line of sight tangent to the surface forms a right triangle at the point on the horizon. Use the leg theorem (or the Pythagorean theorem in similar triangles) to find the straight-line distance $d$ from the hot air balloon to the visible horizon, rounding to two decimal places.
Back to the flagpole: Emma's triangle has legs 5 ft and 4 ft, and the pole's triangle has legs $H$ and 24 ft. They are similar, so $\tfrac{H}{24} = \tfrac{5}{4}$ and the pole is 30 ft tall.