Exercises: Right-angled triangles

In a gable roof that forms a right angle at its apex, the altitude drawn from that vertex divides the base (hypotenuse) into two segments of $2\text{ m}$ and $8\text{ m}$. Apply the altitude theorem to calculate the maximum height of the roof.

A skate ramp has a right-angled triangle profile resting on its hypotenuse. The horizontal base (hypotenuse) measures $10\text{ m}$ and the projection of the upward slope onto the base is $3.6\text{ m}$. Use the cathetus theorem to calculate the length of the upward slope.

A carpenter cuts two wooden wedges in the shape of a right-angled triangle. The first has legs of $15\text{ cm}$ and $20\text{ cm}$. The second has a hypotenuse of $50\text{ cm}$ and a leg of $30\text{ cm}$. Check mathematically whether both wedges are similar.

The two main ropes that secure the central pole of a tent from the top to the ground form a right angle with each other. The pole is $4\text{ m}$ high and one of the ropes is anchored $3.2\text{ m}$ from the base of the pole. At what distance from the base is the other rope anchored?

In a right-angled triangular metal frame, a vertical pillar (altitude to the hypotenuse) divides the bottom beam (hypotenuse) into two parts. If one sloping side (leg) measures $5\text{ m}$ and the total bottom beam measures $10\text{ m}$, how long is the projection of that side onto the bottom beam?

Two right-angled triangular road signs are similar. The shorter leg of the small sign measures $30\text{ cm}$ and its area is $600\text{ cm}^2$. If the shorter leg of the large sign measures $60\text{ cm}$, what is the area of the large sign?

In a technical drawing class, a student draws a right-angled triangle with a hypotenuse of $13\text{ cm}$ and legs of $5\text{ cm}$ and $12\text{ cm}$. Then, they draw the altitude to the hypotenuse. Use the leg theorem to calculate the lengths of both projections of the legs onto the hypotenuse.

You can do it on paper and send a photo with your answer.

Lucas is solving a problem about a ramp with a right-angled triangular cross-section resting on its hypotenuse on a warehouse floor. The statement indicates that the total hypotenuse measures $25\text{ m}$ and that the projection of one of the legs onto that hypotenuse measures $9\text{ m}$. To calculate the height $h$ of the ramp with respect to the ground, Lucas directly applies the geometric mean theorem (altitude theorem) and writes on his exam:

"$h^2 = m \times n \implies h^2 = 25 \times 9 = 225 \implies h = \sqrt{225} = 15\text{ m}$".

Clearly explain what conceptual error Lucas has made when interpreting the data from the statement, write the correct relation he should have applied, and calculate the true height $h$ of the ramp.

In your schoolyard, you want to measure the height of the basketball hoop using the mirror method (based on the fact that the angle of incidence and the angle of reflection of light are identical, generating similar right-angled triangles). You place a horizontal flat mirror on the ground $2.25\text{ m}$ from the base of the hoop post. You walk backwards in a straight line and stand $1.20\text{ m}$ from the mirror, at which point you see the rim accurately reflected. Your eyes are $1.60\text{ m}$ above the ground.

1. a) Before doing any algebraic calculations, observe the relationship between the distance from the mirror to the hoop ($2.25\text{ m}$) and the distance from the mirror to you ($1.20\text{ m}$). Estimate whether the height of the hoop will be greater than, less than, or equal to twice your eye level height ($3.20\text{ m}$), justifying your intuitive estimate.
2. b) Set up the formal similarity proportion between the two right-angled triangles, calculate the exact value of the height of the hoop, and evaluate whether the obtained result confirms your previous estimate and whether it is consistent with the actual dimensions of a regulation basketball hoop.

The façade of a modern building has a right-angled triangular window resting on its hypotenuse. The height of the window relative to the base is $4.8\text{ m}$ and the projection of the shorter side onto the base is $3.6\text{ m}$. Calculate the lengths of both legs and the hypotenuse.

To measure the height of a wind turbine, an engineer places a flat mirror on the ground $12\text{ m}$ from the base of the tower. She steps back $1.5\text{ m}$ from the mirror until she sees the top of the wind turbine reflected exactly in the centre of the mirror. If the engineer's eye level is $1.6\text{ m}$ above the ground, calculate the height of the wind turbine.

A plot of farmland is in the shape of a right-angled triangle. The projection of one leg onto the hypotenuse is $4\text{ dam}$ longer than the projection of the other leg. If the altitude to the hypotenuse is $4.8\text{ dam}$, set up a quadratic equation and calculate the total length of the hypotenuse.

On a billiard table, a ball must hit the cushion and bounce towards a pocket, forming two similar right-angled triangles. The ball is $20\text{ cm}$ from the cushion and the pocket is $60\text{ cm}$ from the same cushion. If the horizontal distance parallel to the cushion between the ball and the pocket is $100\text{ cm}$, at what horizontal distance from the initial position of the ball must the impact on the cushion occur?

In a piece of jewellery, a square of side length $x$ is inscribed inside a right-angled triangle with legs $6\text{ mm}$ and $8\text{ mm}$, such that the vertex of the triangle's right angle coincides with a vertex of the square and the opposite vertex of the square lies on the hypotenuse. Calculate the value of $x$ using similar triangles.

The design of a windsurfing sail is in the shape of a right-angled triangle. It is known that the bottom edge (hypotenuse) measures $5\text{ m}$ and that the leech (one of the legs) measures $1.2\text{ m}$ more than its projection onto the hypotenuse. Determine the two possible lengths for the leech using the leg theorem.

In a museum, a rectangular painting is framed by a right-angled triangle resting on its hypotenuse, such that the altitude to the hypotenuse coincides with one of the sides of the painting. If the projections of the legs onto the base measure $16\text{ cm}$ and $9\text{ cm}$, and the painting occupies exactly the rectangle formed by the altitude and the shorter projection, what is the area of the painting?

A carpentry workshop needs to design a wooden set square in the shape of a right-angled triangle resting on its hypotenuse. The manufacturing specifications impose two conditions: the altitude relative to the hypotenuse must measure exactly $12\text{ cm}$ and the projection of the longer leg onto the hypotenuse must measure $7\text{ cm}$ more than the projection of the shorter leg.

1. a) Set up a quadratic equation using the geometric mean theorem (altitude theorem, $h^2 = m \times n$) to find the value of each of the two projections onto the hypotenuse.
2. b) Solve the equation and obtain the total length of the hypotenuse.
3. c) Apply the leg theorem (geometric mean theorem) to calculate the exact length of both legs and verify, using the global Pythagorean theorem, that the resulting triangle is indeed right-angled.

In an area of the Parque Nacional de Ordesa y Monte Perdido (Huesca), a group of forest rangers needs to estimate the width of an inaccessible ravine through which a torrential river flows. On the opposite bank, there is a vertical beacon $R$ fixed to the rock. To apply the ranging pole method from their accessible bank, they make the following marks in a straight line along the edge:

• They drive a pole $A$ right in front of the beacon $R$, so that the line of sight $RA$ is perpendicular to the bank.

• They walk $15\text{ m}$ along the bank and drive a pole $B$.

• They keep walking $5\text{ m}$ more in the same direction and mark point $C$.

• From $C$, they move inland perpendicularly to the bank until they find a point $D$ from which the pole $B$ and the beacon $R$ are perfectly aligned. They measure the distance traveled: $CD = 8\text{ m}$.

1. a) Formally prove which criterion guarantees that the right-angled triangles $\triangle RAB$ and $\triangle DCB$ are similar.
2. b) Translate the situation into a geometric proportion and calculate the width of the ravine ($RA$).
3. c) If, due to a lack of space on the bank, they had placed pole $B$ $10\text{ m}$ from $A$ and another $10\text{ m}$ from $C$, how many metres would they have had to walk inland from $C$ to align the line of sight? Explain what technical or precision advantage is gained by choosing different distances instead of equal ones.

In a statics study, two vertical poles of heights $a = 120\text{ cm}$ and $b = 80\text{ cm}$ are separated by a distance $d$. From the base of each pole, a tight cable is stretched to the top of the other pole. The cables cross at a height $h$. Prove using similar triangles that the height $h$ at which they cross does not depend on the distance $d$, and calculate its value.

A rectangular sheet of paper $8\text{ cm}$ wide and $15\text{ cm}$ long is folded so that two opposite vertices coincide. The crease formed is a straight line. Calculate the length of this crease based on the similarity of the right-angled triangles formed.

A cable-stayed bridge has a main pylon that forms a right angle with the two longest stay cables. The total length of the deck between the anchors of these cables (hypotenuse) is $125\text{ m}$. If one of the cables measures $25\text{ m}$ more than the other, determine the height of the main pylon (altitude to the hypotenuse).

An industrial designer creates a flat component in the shape of a right-angled triangle whose legs measure $15\text{ cm}$ and $20\text{ cm}$. To reinforce the component, they place a straight rod perpendicular to the hypotenuse passing through the vertex of the right angle (that is, the altitude). In turn, from the foot of this altitude, they run another rod perpendicular to the longer leg. How long is this last rod?

A surveyor wants to calculate the width of a river. To do so, he stands on one bank and sights a tree on the opposite bank. He walks $30\text{ m}$ along the bank (perpendicular to the initial line of sight) and places a marker post. He then walks another $6\text{ m}$ in the same direction, turns $90^\circ$ away from the river, and walks $4\text{ m}$ until he sees the marker post aligned with the tree. What is the width of the river?

Prove that in any right-angled triangle, the reciprocal of the square of the altitude to the hypotenuse is equal to the sum of the reciprocals of the squares of the legs ($\frac{1}{h^2} = \frac{1}{b^2} + \frac{1}{c^2}$). Afterwards, verify this relationship by calculating the altitude in both ways for a triangle with legs $10\text{ cm}$ and $24\text{ cm}$.

A right pyramid with a square base is cut by a plane parallel to the base at one-third of its height (measured from the apex). If the area of the cross-section obtained is $15\text{ cm}^2$, what is the volume of the original pyramid if its total height is $18\text{ cm}$?

In an architectural design, the profile of a roof is a right-angled triangle resting on its hypotenuse. The altitude drawn from the vertex of the right angle divides the hypotenuse into two segments in the ratio $9:16$. If the altitude measures $6\text{ m}$, calculate the total perimeter of the triangle that forms the roof.

To calculate the altitude relative to the hypotenuse ($h$) of a right-angled triangular plot of land with hypotenuse $a = 50\text{ m}$ and leg $b = 30\text{ m}$, two surveyors propose different methods without using trigonometric ratios:

• Method A (Aoife): First, she applies the Pythagorean theorem to calculate the other leg $c$. Then, she equates two expressions for the area of the triangle ($\text{area} = \frac{a \times h}{2}$ and $\text{area} = \frac{b \times c}{2}$) to solve directly for $h = \frac{b \times c}{a}$.

• Method B (Seán): First, he applies the leg theorem ($b^2 = n \times a$) to find the projection $n$. Then, he calculates the other projection $m = a - n$ and, finally, applies the geometric mean theorem (altitude theorem, $h^2 = m \times n$) to solve for $h$.

1. a) Apply both methods detailing the numerical steps and verify that both lead to exactly the same height.
2. b) Analyze the advantages and disadvantages of each method: compare the number of arithmetic operations, the need to calculate square roots, and the risk of propagating errors if the data were non-exact decimals. Which of the two methods would you recommend for operational efficiency? Justify your choice.
3. c) Prove that Aoife's formula ($h = \frac{b \times c}{a}$) is not a numerical coincidence, but a necessary theoretical consequence of the similarity relations $b^2 = n \times a$, $c^2 = m \times a$, and $h^2 = m \times n$, by combining them algebraically.

In a laser cutting workshop, a square opening must be punched inside a sheet of metal in the shape of a right-angled triangle whose legs measure $6\text{ cm}$ and $8\text{ cm}$. The design team is considering two possible configurations to place the square:

• Configuration 1: The square rests on the vertex of the right angle of the triangle, so that two of its sides lie on the legs and the opposite vertex lies on the hypotenuse.

• Configuration 2: One complete side of the square rests on the hypotenuse, and its other two upper vertices each touch a different leg.

1. a) In Configuration 1, identify the similar right-angled triangles delimited by the square inside the main triangle and calculate the exact length of the side of said square ($x_1$).
2. b) In Configuration 2, first calculate the altitude to the hypotenuse of the original triangle. Then, using the similarity between the small upper triangle that crowns the square and the original triangle, set up the corresponding equation and calculate the exact length of the side of the square ($x_2$).
3. c) Mathematically justify which of the two configurations yields a square hole with a larger area and determine the percentage difference in area between the two.