Similarity of right-angled triangles
The altitude drawn from the right angle to the hypotenuse splits a right-angled triangle into two smaller triangles. All three — the large one and the two small ones — are similar to each other. Because the triangles are similar (AA: equal angles, so corresponding sides are in proportion, as in Theorem 13), we get two formulae: the altitude theorem and the leg theorem.
Three similar triangles in one
The altitude theorem
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?
The leg theorem
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?
Applications: roof trusses, sails and bridges
These theorems are not just a geometry exercise: they appear in the design of roof trusses, triangular sails and bridge arches, where a vertical measurement (the "altitude") splits a horizontal measurement (the "hypotenuse") into two sections.
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?
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.
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?
The main sail of a sailboat is in the shape of a right-angled triangle. The seam marking the altitude drawn from the right angle vertex to the hypotenuse measures $6$ m.…
The main sail of a sailboat is in the shape of a right-angled 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 metres.
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.
Solving with an equation
When the information you are given is a relationship between $m$, $n$ and $h$ rather than their values directly, you need to set up an equation using the two theorems and solve it.
If you know the sum $S$ and the product $P$ of two numbers, they are the roots of $x^2 - Sx + P = 0$. For a right-angled triangle, the projections $m$ and $n$ satisfy $m + n = a$ (the hypotenuse) and $m \cdot n = h^2$. So if $m$ and $n$ are the roots of $x^2 - Sx + P = 0$, then $a = S$ and $h = \sqrt{P}$, with no need to solve the equation.
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.