Exercises: 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 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.

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?

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.

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, how many times larger is the area of the large set square compared to the small one?

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?

In a 4th-year ESO Maths exam, Lucía has to calculate the length of the shorter leg, $c$, of a right triangle. The problem states that the projection of that leg onto the hypotenuse measures $m = 3\text{ cm}$ and the projection of the other leg measures $n = 9\text{ cm}$. Lucía writes the following solution:

"By the leg theorem, I know that the square of the leg is the product of the two projections: $c^2 = m \cdot n = 3 \cdot 9 = 27$, therefore $c = \sqrt{27} \approx 5,2\text{ cm}$".

1. a) Explain precisely what conceptual mistake Lucía has made and which theorem or formula she has mixed up.
2. b) Correct the procedure and calculate the exact value of the leg $c$ as a simplified radical and its approximation rounded to the nearest tenth.

In a right triangle, the altitude drawn to the hypotenuse divides it into two segments of lengths $m = 2\text{ cm}$ and $n = 10\text{ cm}$.

1. a) Before carrying out any calculations, reasonably estimate between which two consecutive integers the length of altitude $h$ must lie. Do you think $h$ will be greater than, less than, or equal to half the hypotenuse ($(m+n)/2 = 6\text{ cm}$)? Justify your prediction.
2. b) Apply the corresponding theorem to calculate the exact value of altitude $h$ and obtain its approximation to two decimal places.
3. c) Check whether your calculation confirms the estimate made in part a) and explain why in any non-isosceles right triangle the altitude to the hypotenuse is always strictly less than half of that hypotenuse.

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?

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 $8$ cm, what is the actual height of the tree in metres?

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.

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.

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:

1. a) The longitudinal distance traveled by the ball before the bounce.
2. b) The longitudinal distance traveled after the bounce.

In the Madrid Río park, there are plans to build a skateboard ramp whose side section is a right triangle resting on horizontal ground, which acts as the hypotenuse and measures $3,2\text{ m}$ in length. To reinforce the wooden structure, a vertical steel beam will be placed from the top vertex perpendicular to the ground (acting as the altitude to the hypotenuse). The blueprints indicate that this beam divides the base into two sections, with the longer section being exactly three times the shorter section.

1. a) Model the situation by identifying the geometric elements of the triangle ($a$, $m$, $n$, $h$) and calculate the exact length of each of the two ground sections.
2. b) Calculate the required height of the vertical reinforcing beam and the lengths of the two inclined surfaces of the ramp (the legs). Round your results to the nearest centimetre.
3. c) If the municipal safety regulations for beginner ramps require that the total height does not exceed $1,5\text{ m}$, determine with justification whether the design complies with the regulations.

In a right triangle, we know the exact lengths of its two legs: $b = 15\text{ cm}$ and $c = 20\text{ cm}$. To find the altitude to the hypotenuse, $h$, two students propose different methods:

• Sergio's method: He first calculates the hypotenuse $a$ using the Pythagorean theorem. Then, he uses the geometric mean theorem for legs to find the two projections, $m$ and $n$, onto the hypotenuse. Finally, he calculates $h$ by applying the geometric mean theorem for altitude ($h^2 = m \cdot n$).

• Nerea's method: She first calculates the hypotenuse $a$ using Pythagoras. Next, she equates two ways of expressing the area of the triangle (half the product of the legs versus half the product of the hypotenuse and the altitude: $\frac{b \cdot c}{2} = \frac{a \cdot h}{2}$) to solve for $h$ directly.

1. a) Work through both methods step by step and verify that they yield the exact same result.
2. b) If the legs had non-exact decimal values (for example, $b = 13,4\text{ cm}$ and $c = 18,1\text{ cm}$), which of the two methods would be more advantageous and less prone to accumulating rounding errors? Justify your choice by analysing the number of steps and operations required in each case.

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.

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 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.

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.

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.

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.

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.

Recall that in a monic quadratic equation $x^2 - S\cdot x + P = 0$, 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 ($h^2 = m \cdot n$). 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 right triangle $ABC$ with a right angle at $A$, leg $b$ measures exactly twice leg $c$ ($b = 2c$). The altitude $h$ to the hypotenuse is drawn, dividing it into two segments: $m$ (projection of the shorter leg $c$) and $n$ (projection of the longer leg $b$). This altitude divides the original triangle into two smaller right triangles, $T_1$ and $T_2$.

1. a) Using the geometric mean theorem for legs on both legs, prove deductively that the longer projection is not twice the shorter one, but rather satisfies $n = 4m$.
2. b) Prove, using the geometric mean theorem for altitude, that this altitude measures exactly twice the shorter projection ($h = 2m$).
3. c) A surveying team maps a plot of land with these proportions whose hypotenuse measures a total of $a = 45\text{ m}$. Calculate the measures of $m$, $n$, $h$ and the exact ratio between the area of $T_2$ and the area of $T_1$. Explain how this same area ratio can be deduced directly from triangle similarity without calculating the numerical areas.