Exercises: Similarity of right-angled triangles

In a right triangle, the altitude drawn to the hypotenuse divides it into two segments measuring $4$ in and $9$ in. Calculate the length of this altitude using the corresponding theorem.

The hypotenuse of a right triangle measures $25$ in and the projection of one of its legs onto it measures $9$ in. Calculate the length of this leg applying the leg theorem.

The altitude relative to the hypotenuse of a right triangle measures $6$ in. If one of the segments into which it divides the hypotenuse measures $3$ in, how long is the other segment of the hypotenuse?

A leg of a right triangle measures $8$ in and its orthogonal projection onto the hypotenuse measures $3,2$ in. Determine the total length of the hypotenuse.

Two right triangles are similar. The first has legs measuring $3$ in and $4$ in. If the shorter leg of the second triangle measures $9$ in, how long is the hypotenuse of this second triangle?

A vertical post $2$ ft high casts a shadow of $1,5$ ft on the ground. At the same time and under the same conditions, how long will the shadow cast by a telecommunications tower $40$ ft high be?

Two set squares (which have the shape of a scalene right 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 triangle, the projections of the two legs onto the hypotenuse measure $5$ in and $12$ in, respectively. How long is the hypotenuse of the triangle?

In the design of a truss for a pitched roof in the shape of a right triangle, the vertical beam (which acts as the altitude to the hypotenuse) measures $2,4$ ft and separates the lower beam (hypotenuse) into two sections, one of which measures $1,8$ ft. Calculate the total perimeter of the outer structure of the truss.

A street lamp $4,5$ ft high illuminates a person $1,8$ ft tall who is standing $3$ ft from the base of the lamp. Use the similarity of right triangles to find out the length of the shadow the person casts on the ground.

To measure the height of a building, a student places a flat mirror on the ground $15$ ft from the base of the building. Stepping back $1,2$ ft from the mirror, they see the top of the building reflected in it. If the student's eyes are $1,6$ ft from the ground, what is the height of the building?

An accessibility ramp has a total length of $12$ ft resting on the ground and overcomes a maximum height difference of $1,5$ ft. If a wheelchair has advanced $4$ ft up the ramp from its start at the bottom, at what height is it relative to ground level?

In a homemade pinhole camera $20$ in deep, an inverted image of a tree located $15$ ft from the pinhole is formed. If the image formed at the back of the camera has a height of $8$ in, what is the real height of the tree in feet?

The main sail of a sailboat has the shape of a right triangle. The seam that marks the altitude drawn from the vertex of the right angle to the hypotenuse measures $6$ ft. If the projection of the shorter side of the sail onto the hypotenuse is $4,5$ ft, calculate the total area of the sail in square feet.

A stone bridge arch has the shape of a perfect semicircle with a diameter of $20$ ft. At $4$ ft from one of the ends of the bridge, a vertical pillar needs to be placed reaching from the ground to the arch. Using the altitude theorem, determine the exact length this pillar must have.

On a rectangular billiard table, a ball is hit without spin against the opposite cushion, forming two similar right triangles in its path. The ball starts at $40$ in from the side cushion and rebounds at $60$ in from the same cushion. If the total longitudinal distance the ball has advanced along the table is $150$ in, determine the longitudinal distances traveled before and after the rebound.

In a plot of land shaped like a right triangle, an irrigation trench has been dug that coincides with the altitude to the hypotenuse and measures $12$ ft. If the ratio between the two segments into which this trench divides the hypotenuse is $\frac{9}{16}$, calculate the dimensions of the three sides that bound the land.

In a right triangle, the projection of the shorter leg onto the hypotenuse measures $x$ in, the projection of the longer leg measures $(x+5)$ in, and the altitude relative to the hypotenuse measures $(x+2)$ in. Set up the corresponding equation, find the value of $x$, and calculate the total area of the triangle.

Two vertical posts of heights $a = 3$ ft and $b = 6$ ft are separated by an unknown distance. Two taut cables are stretched from the base of each post to the top of the other post. Using the similarity of the right triangles formed, show that the height $h$ at which the cables cross is independent of the distance separating the posts and calculate its value.

A square is inscribed in a right triangle with legs of $8$ in and $6$ in, such that one of its vertices coincides with the right angle of the triangle, and the two sides of the square forming that vertex rest on the legs. Calculate the measure of the side of the square $x$ using the similarity of the right triangles formed.

The area of a solar panel shaped like a right triangle is $150$ in$^2$ and the altitude relative to its hypotenuse measures $12$ in. 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 triangle, the ratio between the squares of the legs is $\frac{b^2}{c^2} = \frac{9}{16}$. If the hypotenuse measures $50$ in, use the proportionality relationship deduced from the leg theorems to first find the measure of the projections of both legs onto the hypotenuse, and subsequently the exact length of the legs.

A conical glass has a total depth of $15$ in and a radius of $6$ in at its top opening. It is filled with water until the liquid reaches a height of $10$ in measured from the bottom (lower vertex). Calculate the radius of the circular surface of the water using the similarity of right triangles formed with the central axis of the glass.

If we consider the Earth to be a sphere of radius $R = 6371$ miles, and a person is in a hot air balloon at a height of $h = 2$ miles above the surface, the line of sight tangent to the surface forms a right triangle at the point on the horizon. Use the leg theorem (or the Pythagorean 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.