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 center pole of a tent forms a right angle with the two main ropes securing it from the top to the ground. The pole is $4\text{ m}$ high and one of the ropes is anchored $3.2\text{ m}$ from the base of the pole. How far 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.

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 center 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?

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 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.