Before you start, make sure you know: Pythagorean Theorem (Secondary Year 2 (first cycle))
Before you start, make sure you know: Similarity of Triangles (Secondary Year 2 (first cycle))
Wilkin wants to know how wide a river is, but he can't cross it with a tape measure. With a couple of stakes and a little similarity he does it without getting his shoes wet. The altitude and leg theorems are the basis of techniques like this one. In this section you prove them and apply them to real situations.
Quick review: altitude and leg
Why do they hold? The altitude splits the triangle into two smaller right triangles. Let's see that all three are similar and compare their sides.
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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.
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)…
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?
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…
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.
Two right triangles already share the right angle. So they are similar if both legs are proportional, or if a leg and the hypotenuse are proportional. If you are missing a side, you find it with Pythagoras.
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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 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…
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.
Sometimes you aren't given the projections, but a relationship between them (one is larger than the other, or they are in proportion). Then you call one projection $x$, write the other in terms of $x$, and apply the altitude or leg theorem to set up an equation.
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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…
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.
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…
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.
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…
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).
A flat mirror on the ground obeys the rule that the angle of incidence equals the angle of reflection. This creates two similar right triangles (the one with your eyes and the mirror, and the one with the object and the mirror), which let you calculate heights without climbing anything.
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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…
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.
Overlapping triangles: billiards and inscribed squares
When a figure hides two right triangles that share an angle, they are similar, and the proportion between their sides solves problems that at first glance don't look like similarity. Think of a ball bouncing off the cushion of a pool table.
Inscribed square
If you inscribe a square of side $x$ in a right triangle, with one vertex at the right angle and the opposite vertex on the hypotenuse, the square leaves a small triangle on one side that is similar to the original. The legs of that small triangle are $x$ (the side of the square) and what is left of the other leg of the large triangle after removing the square (leg minus $x$). You set it up with the proportion between those legs and the legs of the large triangle.
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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.
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…
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.
To measure the width of a river without crossing it, you use stakes (or rods) to build two similar right triangles on one bank: a large one, reaching the tree on the other bank, and a small one that you can measure entirely.
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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?
If you cut a pyramid with a plane parallel to the base, the top part is another pyramid similar to the original. Its lengths are in ratio $k$, its areas in ratio $k^2$ and its volumes in ratio $k^3$.
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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?
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…
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 $18\text{ cm}$?