Right Triangles
The altitude theorem and the leg theorem aren't just formulas to memorize: they are the basis of real techniques for measuring things you can't measure with a tape measure, from the height of a wind turbine in La Ventosa, Oaxaca, to the width of a river. They all come from the same idea, the similarity of triangles, and in this section you will apply them to real problems.
Quick review: altitude and leg
You don't have to memorize them blindly: the altitude splits the large triangle into two smaller right triangles, and all three are similar (they share angles). Each formula is a proportion between corresponding sides of two of them, written without fractions.
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…
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.
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…
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?
Checking whether two right triangles are similar
For two right triangles to be similar, it is enough that their two legs are proportional, or that a leg and the hypotenuse are proportional: they already share the right angle.
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…
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…
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?
The mirror method
A flat mirror on the ground reflects angles the same way they are produced: the angle of incidence equals the angle of reflection. This creates two similar right triangles (the one formed by your eyes and the mirror, and the one formed by the object and the mirror) that let you calculate unreachable heights without climbing anything.
To measure the height of a wind turbine, an engineer places a plane mirror on the ground $12\text{ m}$ from the base of the tower. She steps back $1.5\text{ m}$ from the…
To measure the height of a wind turbine, an engineer places a plane 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.
Overlapping triangles: billiards and inscribed squares
When a figure hides two right triangles that share an angle (and are therefore similar), the proportion between their sides solves problems that at first glance don't look like similarity problems.
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…
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…
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.
Measuring an unreachable width: the stake method
To measure the width of a river without crossing it, you use two similar right triangles built with stakes (posts or rods driven into the ground as markers) on one bank: a large one (reaching the tree on the other bank) and a small one that you can measure entirely.
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…
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?
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…
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.