Similarity of right-angled triangles

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 other segment of the hypotenuse?

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.

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-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 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 metres?

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 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 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. 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 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.

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.

Recall that in a monic quadratic equation $x^2 - S\times x + P = 0$, the coefficient $S$ represents the sum of its two solutions and the constant term $P$ represents their product.

1. a) In a right triangle, the entire hypotenuse is the sum of the projections ($a = m + n$) and the square of the altitude to the hypotenuse is the product of these projections ($h^2 = m \times n$). Without solving the equation, deduce the value of the hypotenuse $a$ and that of the altitude $h$ of a right triangle, knowing that the measures of its projections $m$ and $n$ are the roots of the equation $x^2 - 13x + 36 = 0$ (with measurements in centimetres).
2. b) Pose the inverse problem: write the quadratic equation whose solutions correspond to the projections $m$ and $n$ of a right triangle with a hypotenuse of $a = 25\text{ cm}$ and an altitude to the hypotenuse of $h = 12\text{ cm}$.
3. c) Solve the equation you formulated in part b) to find $m$ and $n$, and use them to calculate the exact lengths of the two legs of the triangle.

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 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 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.