Exercises: Right-angled triangles

In a pitched roof that forms a right angle at its top vertex, the altitude drawn from this 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 the profile of a right triangle resting on its hypotenuse. The horizontal base (hypotenuse) measures $10\text{ m}$ and the projection of the upward section onto the base is $3,6\text{ m}$. Use the leg theorem to calculate the length of the upward section.

A carpenter cuts two wooden wedges in the shape of a right 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 $30\text{ cm}$. Mathematically check if both wedges are similar.

The central mast of a tent forms a right angle with the two main ropes that secure it from the top to the ground. The mast is $4\text{ m}$ high and one of the ropes is anchored at $3,2\text{ m}$ from the base of the mast. At what distance from the base is the other rope anchored?

A volleyball post casts a shadow of $3,5\text{ m}$ on the horizontal ground. At the same time and in the same place, a water bottle $0,25\text{ m}$ high casts a shadow of $0,35\text{ m}$. Calculate the height of the post using similarity of triangles.

In a metal structure shaped like a right triangle, a vertical pillar (altitude to the hypotenuse) divides the lower beam (hypotenuse) into two parts. If an inclined side (leg) measures $5\text{ m}$ and the total lower beam measures $10\text{ m}$, how long is the projection of that side onto the lower beam?

Two traffic signs shaped like right triangles 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 triangle with a hypotenuse of $13\text{ cm}$ and legs of $5\text{ cm}$ and $12\text{ cm}$. Then, they draw the altitude relative to the hypotenuse. Use the leg theorem to calculate the lengths of the two projections of the legs onto the hypotenuse.

The facade of a modern building has a large window shaped like a right triangle resting on its hypotenuse. The height of the window with respect 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 length of the two 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 highest point of the wind turbine reflected exactly in the center of the mirror. If the engineer's eyes are $1,6\text{ m}$ from the ground, calculate the height of the wind turbine.

An agricultural plot of land has the shape of a right triangle. The projection of one of the legs onto the hypotenuse is $4\text{ dam}$ greater than the projection of the other leg. If the altitude relative to the hypotenuse measures $4,8\text{ dam}$, set up a quadratic equation and calculate the total length of the hypotenuse.

On a billiard table, a ball must hit the cushion and rebound towards a pocket, forming two similar right 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 ball's initial position must the impact on the cushion occur?

An opaque fence $2\text{ m}$ high is located in front of the vertical wall of a building. A spotlight on the ground $1,5\text{ m}$ from the fence casts its shadow onto the building, reaching a height of $6\text{ m}$. If we move the spotlight $1\text{ m}$ further from the fence (placing it at $2,5\text{ m}$), how much will the height of the shadow cast on the building decrease?

In a piece of jewelry, a square of side $x$ is inscribed within a right triangle with legs of $6\text{ mm}$ and $8\text{ mm}$, such that the vertex of the right angle of the triangle coincides with a vertex of the square and the opposite vertex of the square rests on the hypotenuse. Calculate the value of $x$ using similarity of triangles.

The design of a windsurf sail has the shape of a right triangle. It is known that the lower edge (hypotenuse) measures $5\text{ m}$ and that the trailing edge (one of the legs) measures $1,2\text{ m}$ more than its projection onto the hypotenuse. Determine the two possible lengths the trailing edge can have using the leg theorem.

In a museum, a rectangular painting is framed by a right 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 posts of heights $a = 120\text{ cm}$ and $b = 80\text{ cm}$ are separated by a distance $d$. From the base of each post, a taut cable is pulled to the top of the other post. The cables cross at a height $h$. Show using similarity of 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 fold formed is a straight line. Calculate the length of this fold based on the similarity of the right triangles formed.

A cable-stayed bridge has a main pillar that forms a right angle with the two longest stay cables. The total length of the deck between the anchorages 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 pillar (altitude to the hypotenuse).

An industrial designer creates a flat part in the shape of a right triangle whose legs measure $15\text{ in}$ and $20\text{ in}$. To reinforce the part, 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 place 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 this, they stand on one bank and observe a tree on the opposite bank. They walk $30\text{ m}$ along the bank (perpendicular to the initial line of sight) and place a ranging pole. Then they walk another $6\text{ m}$ in the same direction, turn $90^\circ$ away from the river and walk $4\text{ m}$ until they see the pole aligned with the tree. What is the width of the river?

Prove that in any right 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 ($\frac{1}{h^2} = \frac{1}{b^2} + \frac{1}{c^2}$). Then, verify this relationship by calculating the altitude in both ways in a triangle with legs $10\text{ cm}$ and $24\text{ cm}$.

A right square-based pyramid is cut by a plane parallel to the base at one third of its height (measuring from the vertex). 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 triangle resting on its hypotenuse. The altitude drawn from the vertex of the right angle divides the hypotenuse into two segments that are in the ratio $9:16$. If the altitude measures $6\text{ m}$, calculate the total perimeter of the triangle forming the roof.