Before you start, make sure you know: Pythagorean Theorem (Secondary Year 2 (first cycle))
Yaritza wants to know how tall a building in Santiago is without climbing it. All she has is a tape measure and a little mirror from her purse. Can she do it? Yes, because there are similar triangles hidden inside a right triangle. At the end of the page we solve it.
Three similar triangles in one
If you draw the altitude from the right angle to the hypotenuse of a right triangle, the triangle is split into two smaller triangles. All three (the big one and the two small ones) are similar to each other.
In triangle $ABC$ the right angle is at $A$ and the hypotenuse is $\overline{BC}$. Point $H$ is the foot of the altitude $h = \overline{AH}$. The projections of the legs onto the hypotenuse are $m = \overline{BH}$ and $n = \overline{HC}$. Leg $c = \overline{AB}$ projects onto $m$ and leg $b = \overline{AC}$ projects onto $n$.
The altitude theorem
A very visual case is the arch of a bridge. An angle inscribed in a semicircle is a right angle. So if you join a point on the arch to the two ends of the diameter, you form a right triangle. The height of that point above the diameter is the altitude to the hypotenuse.
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In a right-angled triangle, the altitude to the hypotenuse divides it into two segments measuring $4$ cm and $9$ cm. Calculate the length of this altitude using the…
In a right-angled triangle, the altitude to the hypotenuse divides it into two segments measuring $4$ cm and $9$ cm. Calculate the length of this altitude using the appropriate theorem.
The altitude to the hypotenuse of a right-angled triangle measures $6$ cm. If one of the segments into which it divides the hypotenuse measures $3$ cm, how long is the…
The altitude to the hypotenuse of a right-angled triangle measures $6$ cm. If one of the segments into which it divides the hypotenuse measures $3$ cm, how long is the other segment of the hypotenuse?
A stone bridge arch has the shape of a perfect semicircle with a diameter of $20$ m. At $4$ m from one end of the bridge, a vertical pillar must be placed reaching from…
A stone bridge arch has the shape of a perfect semicircle with a diameter of $20$ m. At $4$ m from one end of the bridge, a vertical pillar must be placed reaching from the ground to the arch. Using the altitude theorem, determine the exact length this pillar must have.
The hypotenuse of a right-angled triangle measures $25$ cm and the projection of one of its legs onto it measures $9$ cm. Calculate the length of this leg using the…
The hypotenuse of a right-angled triangle measures $25$ cm and the projection of one of its legs onto it measures $9$ cm. Calculate the length of this leg using the cathetus theorem.
One leg of a right-angled triangle measures $8$ cm and its orthogonal projection onto the hypotenuse measures $3.2$ cm. Determine the total length of the hypotenuse.
One leg of a right-angled triangle measures $8$ cm and its orthogonal projection onto the hypotenuse measures $3.2$ cm. Determine the total length of the hypotenuse.
In a right-angled triangle, the projections of the two legs onto the hypotenuse measure $5$ cm and $12$ cm respectively. What is the length of the hypotenuse of the…
In a right-angled triangle, the projections of the two legs onto the hypotenuse measure $5$ cm and $12$ cm respectively. What is the length of the hypotenuse of the triangle?
Two right triangles with an equal acute angle are similar (AA). Then all their corresponding sides have the same ratio $k$. But the areas have ratio $k^2$.
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Two right-angled triangles are similar. The legs of the first measure $3$ cm and $4$ cm. If the shorter leg of the second triangle measures $9$ cm, what is the length of…
Two right-angled triangles are similar. The legs of the first measure $3$ cm and $4$ cm. If the shorter leg of the second triangle measures $9$ cm, what is the length of the hypotenuse of this second triangle?
Two set squares (in the shape of a scalene right-angled triangle) are similar. If the hypotenuse of the larger set square is three times that of the smaller set square,…
Two set squares (in the shape of a scalene right-angled 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?
Zinc roofs rest on trusses. The sails of sailboats and solar panels are also often right triangles. In all of them, a vertical measurement (the altitude) splits the slanted base (the hypotenuse) into two sections.
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In the design of a roof truss for a gable roof in the shape of a right-angled triangle, the vertical beam (which acts as the altitude to the hypotenuse) measures $2.4$ m…
In the design of a roof truss for a gable roof in the shape of a right-angled triangle, the vertical beam (which acts as the altitude to the hypotenuse) measures $2.4$ m and divides the bottom beam (hypotenuse) into two sections, one of which measures $1.8$ m. Calculate the total perimeter of the outer structure of the truss.
The main sail of a sailboat is in the shape of a right triangle. The seam marking the altitude drawn from the right angle vertex to the hypotenuse measures $6$ m. If the…
The main sail of a sailboat is in the shape of a right triangle. The seam marking the altitude drawn from the right angle vertex to the hypotenuse measures $6$ m. If the projection of the shorter side of the sail onto the hypotenuse is $4.5$ m, calculate the total area of the sail in square meters.
Sometimes the given information is not a direct value, but a relationship between $m$, $n$, and $h$ (a ratio, a difference, a sum). Then you write the projections using an unknown, apply the theorem, and solve.
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In a plot of land shaped like a right-angled triangle, an irrigation ditch coinciding with the altitude to the hypotenuse has been dug and measures $12$ m. If the ratio…
In a plot of land shaped like a right-angled triangle, an irrigation ditch coinciding with the altitude to the hypotenuse has been dug and measures $12$ m. If the ratio between the two segments into which this ditch divides the hypotenuse is $\frac{9}{16}$, calculate the dimensions of the three sides bounding the land.
In a right-angled triangle, the projection of the shorter leg onto the hypotenuse measures $x$ cm, the projection of the longer leg measures $(x+5)$ cm, and the altitude…
In a right-angled triangle, the projection of the shorter leg onto the hypotenuse measures $x$ cm, the projection of the longer leg measures $(x+5)$ cm, and the altitude to the hypotenuse measures $(x+2)$ cm. Set up the corresponding equation, find the value of $x$, and calculate the total area of the triangle.
In a right-angled triangle, the ratio between the squares of the legs is $\frac{b^2}{c^2} = \frac{9}{16}$. If the hypotenuse measures $50$ cm, use the proportionality…
In a right-angled triangle, the ratio between the squares of the legs is $\frac{b^2}{c^2} = \frac{9}{16}$. If the hypotenuse measures $50$ cm, use the proportionality relationship deduced from the cathetus theorems to find first the lengths of the projections of both legs onto the hypotenuse, and then the exact length of the legs.
Other times the given information is the area and the altitude. With them you know two things about the projections: their sum (the hypotenuse) and their product (which is $h^2$). That is a system of equations.
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The area of a solar panel shaped like a right-angled triangle is $150$ dm$^2$ and the altitude to its hypotenuse measures $12$ dm. By setting up a system of equations…
The area of a solar panel shaped like a right-angled triangle is $150$ dm$^2$ and the altitude to its hypotenuse measures $12$ dm. 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.
Measuring with similarity: mirror, pole, ramps, and camera obscura
You don't need the altitude theorem to measure things: it is enough to find two similar right triangles and set up a proportion. This is the method used by surveyors and bricklayers, who use a mirror on the ground or a range pole (a stake or rod driven into the ground) to measure heights and distances without climbing or crossing anything.
Light reflects at the same angle at which it arrives. So the triangles of the student and of the building have an equal acute angle and a right angle: they are similar by AA.
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To measure the height of a building, a student places a flat mirror on the ground $15$ m from the base of the building. Stepping back $1.2$ m from the mirror, they see…
To measure the height of a building, a student places a flat mirror on the ground $15$ m from the base of the building. Stepping back $1.2$ m from the mirror, they see the top of the building reflected in it. If the student's eyes are $1.6$ m from the ground, what is the height of the building?
An accessibility ramp has a total length of $12$ m resting on the ground and covers a maximum vertical rise of $1.5$ m. If a wheelchair has travelled $4$ m along the…
An accessibility ramp has a total length of $12$ m resting on the ground and covers a maximum vertical rise of $1.5$ m. If a wheelchair has travelled $4$ m along the ramp from the bottom, at what height is it above ground level?
In a homemade pinhole camera $20$ cm deep, the inverted image of a tree $15$ m from the pinhole is formed. If the image formed at the back of the camera has a height of…
In a homemade pinhole camera $20$ cm deep, the inverted image of a tree $15$ m from the pinhole is formed. If the image formed at the back of the camera has a height of $8$ cm, what is the actual height of the tree in meters?
The same idea works when the triangles are "hidden": a bounce, a conical cup, two cables that cross. You always look for equal angles and write the proportion of corresponding sides.
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On a rectangular pool table, you hit a ball without spin against one of the cushions, so that it bounces off forming two similar right-angled triangles in its path. The…
On a rectangular pool table, you hit a ball without spin against one of the cushions, so that it bounces off forming two similar right-angled triangles in its path. The ball starts from a position located $40\text{ cm}$ from that cushion and, after the bounce, reaches a point located $60\text{ cm}$ from the same cushion. If the total longitudinal distance covered by the ball in the direction parallel to the cushion is $150\text{ cm}$, determine:
a) The longitudinal distance traveled by the ball before the bounce.
b) The longitudinal distance traveled after the bounce.
A conical glass tumbler has a total depth of $15$ cm and a top opening radius of $6$ cm. It is filled with water until the liquid reaches a height of $10$ cm measured…
A conical glass tumbler has a total depth of $15$ cm and a top opening radius of $6$ cm. It is filled with water until the liquid reaches a height of $10$ cm measured from the bottom (lower vertex). Calculate the radius of the circular surface of the water using the similarity of the right-angled triangles formed with the central axis of the glass.
Two vertical poles of heights $a = 3$ m and $b = 6$ m are separated by an unknown distance. Two taut cables are stretched from the base of each pole to the top end of…
Two vertical poles of heights $a = 3$ m and $b = 6$ m are separated by an unknown distance. Two taut cables are stretched from the base of each pole to the top end of the other pole. Using the similarity of the right-angled triangles formed, prove that the height $h$ at which the cables cross is independent of the distance separating the poles, and calculate its value.
In a right-angled triangle with legs $8$ cm and $6$ cm, a square is inscribed such that one of its vertices coincides with the right angle of the triangle, and the two…
In a right-angled triangle with legs $8$ cm and $6$ cm, a square is inscribed such that one of its vertices coincides with the right angle of the triangle, and the two sides of the square forming that vertex lie on the legs. Calculate the side length of the square $x$ using the similarity of the right-angled triangles formed.
If we consider the Earth to be a sphere of radius $R = 6371$ km, and a person is in a hot air balloon at a height of $h = 2$ km above the surface, the line of sight…
If we consider the Earth to be a sphere of radius $R = 6371$ km, and a person is in a hot air balloon at a height of $h = 2$ km above the surface, the line of sight tangent to the surface forms a right-angled triangle at the horizon point. Use the cathetus theorem (or Pythagoras' 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.