Similarity of Right Triangles
The altitude drawn from the right angle to the hypotenuse splits a right triangle into two smaller triangles. All three —the large one and the two small ones— are similar to each other. Two formulas come from this that you will use constantly: the altitude theorem and the leg theorem.
Three similar triangles in one
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?
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 arches
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?
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-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 meters.
Solving with an equation
When the given information is a relationship among $m$, $n$, and $h$ rather than their direct values, you need to set up an equation using the two theorems and solve it.
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.