Before you start, make sure you know: Quadratic equations (Math 8 (Pre-Algebra))
Your family just bought a "55-inch" TV. Nobody measured 55 inches along the bottom, because the 55 is the length of the screen's diagonal. If the screen is 48 inches wide and 27 inches tall, is the label honest? A right triangle is hiding in that screen, and the Pythagorean Theorem will find it.
You already know the theorem from Math 7 and 8. In Geometry you use it much more flexibly: for the apothem of a hexagon, the diagonal of a box, or an equation when several lengths are unknown at once.
The Pythagorean Theorem
The hypotenuse is the side opposite the right angle, and it is always the longest side. The other two sides are the legs.
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A fire truck ladder leans against the facade of a building. The base of the ladder is $3\text{ m}$ from the wall and reaches a height of $4\text{ m}$ on the facade.…
A fire truck ladder leans against the facade of a building. The base of the ladder is $3\text{ m}$ from the wall and reaches a height of $4\text{ m}$ on the facade. Calculate the total length of the ladder.
A telecommunications mast is supported by a taut cable from its top end to a point on the ground. The cable measures $13\text{ m}$ and the mast has a height of…
A telecommunications mast is supported by a taut cable from its top end to a point on the ground. The cable measures $13\text{ m}$ and the mast has a height of $12\text{ m}$. Determine at what distance from the base of the mast the cable is anchored to the ground.
A vertical antenna is supported by two cables anchored to the ground on opposite sides of the base. One cable measures $15\text{ m}$ and is anchored $9\text{ m}$ from…
A vertical antenna is supported by two cables anchored to the ground on opposite sides of the base. One cable measures $15\text{ m}$ and is anchored $9\text{ m}$ from the base. The other cable is anchored $16\text{ m}$ from the base. Calculate the height of the antenna and the length of the second cable.
A diagonal cuts a rectangle into two right triangles. The sides of the rectangle are the legs, and the diagonal is the hypotenuse.
Back to the TV: the "55 inches" is the diagonal of the screen, and the Pythagorean Theorem confirms it.
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The dimensions of a rectangular computer screen are $32\text{ cm}$ wide and $24\text{ cm}$ high. Calculate the length of its diagonal to find out the actual size of the…
The dimensions of a rectangular computer screen are $32\text{ cm}$ wide and $24\text{ cm}$ high. Calculate the length of its diagonal to find out the actual size of the screen.
A town square is shaped like a square with a side of $50\text{ m}$. If you decide to cross it in a straight line from one corner to the opposite corner, what distance…
A town square is shaped like a square with a side of $50\text{ m}$. If you decide to cross it in a straight line from one corner to the opposite corner, what distance will you cover?
The diagonals of a rhombus are perpendicular bisectors of each other. So each side of a rhombus is the hypotenuse of a right triangle whose legs are half of each diagonal.
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To build a kite in the shape of a rhombus, two wooden rods measuring $60\text{ cm}$ and $80\text{ cm}$ have been crossed. Calculate the length of the outer edge of the…
To build a kite in the shape of a rhombus, two wooden rods measuring $60\text{ cm}$ and $80\text{ cm}$ have been crossed. Calculate the length of the outer edge of the kite (the side of the rhombus).
The perimeter of a rhombus-shaped swimming pool is $52\text{ m}$ and one of its diagonals measures $10\text{ m}$. Calculate the length of the other diagonal and the…
The perimeter of a rhombus-shaped swimming pool is $52\text{ m}$ and one of its diagonals measures $10\text{ m}$. Calculate the length of the other diagonal and the surface area of the pool.
When a figure has no right angle, make one. Draw an altitude, and the Pythagorean Theorem has a triangle to work with.
In an isosceles triangle, the altitude to the base also bisects the base. That is why the legs of the right triangle are $h$ and half the base.
A right trapezoid works the same way. Drop a perpendicular from the end of the shorter base. The slanted side is the hypotenuse of a triangle whose legs are the height and the difference of the bases.
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A traffic sign is in the shape of an isosceles triangle. Its base measures $60\text{ cm}$ and each of the two equal sides measures $50\text{ cm}$. Calculate the height…
A traffic sign is in the shape of an isosceles triangle. Its base measures $60\text{ cm}$ and each of the two equal sides measures $50\text{ cm}$. Calculate the height $h$ of the sign.
The cross-section of an irrigation canal is in the shape of an isosceles trapezoid. The width at the top is $8\text{ m}$, the width at the bottom is $2\text{ m}$ and the…
The cross-section of an irrigation canal is in the shape of an isosceles trapezoid. The width at the top is $8\text{ m}$, the width at the bottom is $2\text{ m}$ and the depth of the canal is $4\text{ m}$. Calculate the length of the sloping sides $x$ of the canal.
A plot of land is in the shape of a right trapezoid. The side perpendicular to the bases measures $12\text{ m}$, the longer base measures $30\text{ m}$ and the shorter…
A plot of land is in the shape of a right trapezoid. The side perpendicular to the bases measures $12\text{ m}$, the longer base measures $30\text{ m}$ and the shorter base measures $25\text{ m}$. Calculate the total perimeter of the plot.
The apothem of a regular polygon is the segment from the center perpendicular to a side. The radius goes from the center to a vertex. Together with half a side, they form a right triangle.
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A decorative mosaic is made up of tiles in the shape of a regular hexagon with a side of $10\text{ cm}$. Calculate the apothem of one of these tiles.
A decorative mosaic is made up of tiles in the shape of a regular hexagon with a side of $10\text{ cm}$. Calculate the apothem of one of these tiles.
A radius that is perpendicular to a chord bisects the chord. So the radius, half the chord, and the distance from the center to the chord form a right triangle, with the radius as hypotenuse.
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A cylindrical pipe carries water. The radius of the pipe is $13\text{ cm}$ and the surface of the water forms a horizontal chord $24\text{ cm}$ wide. If the water level…
A cylindrical pipe carries water. The radius of the pipe is $13\text{ cm}$ and the surface of the water forms a horizontal chord $24\text{ cm}$ wide. If the water level is below the center of the pipe, what is the maximum depth of the water $h$?
In a circular piece of wood with a radius of $25\text{ cm}$, two straight and parallel cuts are made. One of the cuts has a length of $48\text{ cm}$ and the other…
In a circular piece of wood with a radius of $25\text{ cm}$, two straight and parallel cuts are made. One of the cuts has a length of $48\text{ cm}$ and the other $40\text{ cm}$. If both cuts are made on opposite sides of the center of the circle, calculate the perpendicular distance separating the two cuts.
A logo is formed by an isosceles triangle inscribed in a circle. The base of the triangle measures $16\text{ cm}$ and its altitude to the base measures $18\text{ cm}$.…
A logo is formed by an isosceles triangle inscribed in a circle. The base of the triangle measures $16\text{ cm}$ and its altitude to the base measures $18\text{ cm}$. Calculate the radius of the circle in which the triangle is inscribed.
Draw the altitude from the right angle to the hypotenuse. It splits the triangle into two smaller right triangles. Both are similar to the original triangle (AA) and to each other. That similarity gives a shortcut that is a cousin of the Pythagorean Theorem.
The geometric mean of two positive numbers $x$ and $y$ is $\sqrt{xy}$.
Many problems just call this one the altitude theorem and the next one the leg theorem.
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In the design of the roof of an industrial building, the main structure forms a right triangle. The altitude to the hypotenuse divides it into two segments of…
In the design of the roof of an industrial building, the main structure forms a right triangle. The altitude to the hypotenuse divides it into two segments of $4\text{ m}$ and $9\text{ m}$. Apply the altitude theorem to calculate the length of this altitude.
The roof of a house is in the shape of a right triangle resting on its hypotenuse. The height of the roof with respect to the horizontal base is $6\text{ m}$ and the…
The roof of a house is in the shape of a right triangle resting on its hypotenuse. The height of the roof with respect to the horizontal base is $6\text{ m}$ and the projection of one of the slopes (leg) onto the base is $8\text{ m}$. Calculate the length of the other projection and the measure of the two sloping sides of the roof.
Each leg also has a similar-triangle relationship with the hypotenuse. Look at the figure in the previous section. The small triangle with leg $a$ and segment $p$ shares an angle with the big triangle, so they are similar.
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In a park, a flowerbed is in the shape of a right triangle. The hypotenuse measures $25\text{ m}$ and the projection of one of the legs onto the hypotenuse measures…
In a park, a flowerbed is in the shape of a right triangle. The hypotenuse measures $25\text{ m}$ and the projection of one of the legs onto the hypotenuse measures $9\text{ m}$. Use the leg theorem to calculate the length of that leg.
In a right triangle, the altitude to the hypotenuse measures $12\text{ cm}$ and one of the legs measures $15\text{ cm}$. Using the leg and altitude theorems, or the…
In a right triangle, the altitude to the hypotenuse measures $12\text{ cm}$ and one of the legs measures $15\text{ cm}$. Using the leg and altitude theorems, or the Pythagorean theorem iteratively, calculate the length of the hypotenuse $a$ and the other leg $c$.
In three dimensions, apply the theorem twice in a row. First use a flat right triangle in a face to get a diagonal. Then combine that diagonal with a perpendicular edge.
The purple dashed segment is the diagonal of the base. The blue segment is the space diagonal.
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To transport a rigid metal bar, a box in the shape of a rectangular prism is available, whose dimensions are $80\text{ cm}$ long, $60\text{ cm}$ wide and $50\text{ cm}$…
To transport a rigid metal bar, a box in the shape of a rectangular prism is available, whose dimensions are $80\text{ cm}$ long, $60\text{ cm}$ wide and $50\text{ cm}$ high. Calculate the maximum length the bar can have to fit completely enclosed inside the box.
The Great Pyramid of Giza has a square base. Suppose that the side of its base measures $230\text{ m}$ and that the length of one of its lateral edges (from a vertex of…
The Great Pyramid of Giza has a square base. Suppose that the side of its base measures $230\text{ m}$ and that the length of one of its lateral edges (from a vertex of the base to the apex) is $219\text{ m}$. Calculate the height $h$ of the pyramid.
A room is in the shape of a rectangular prism with dimensions $5\text{ m}$ long, $4\text{ m}$ wide and $3\text{ m}$ high. A spider is in one corner of the ceiling and…
A room is in the shape of a rectangular prism with dimensions $5\text{ m}$ long, $4\text{ m}$ wide and $3\text{ m}$ high. A spider is in one corner of the ceiling and wants to reach the opposite corner of the floor by walking along the walls, ceiling or floor. If the spider takes the shortest possible path, calculate this distance. (Hint: imagine the net of the room).
Sometimes no side is fully known. You only know how the sides relate. Name the unknowns, apply the theorem, and solve the equation. This often means solving a quadratic equation, or a system when two unknowns appear.
For consecutive even integers, use $x$, $x+2$, $x+4$ instead.
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In a garden, an area in the shape of a right triangle is to be marked out. It is known that the lengths of its three sides are three consecutive even integers. Set up…
In a garden, an area in the shape of a right triangle is to be marked out. It is known that the lengths of its three sides are three consecutive even integers. Set up the corresponding equation using the Pythagorean theorem and determine the lengths of the three sides.
A bridge has a lateral structure in the shape of a scalene trapezoid. The horizontal bases measure $40\text{ m}$ and $26\text{ m}$, and the oblique sides measure…
A bridge has a lateral structure in the shape of a scalene trapezoid. The horizontal bases measure $40\text{ m}$ and $26\text{ m}$, and the oblique sides measure $13\text{ m}$ and $15\text{ m}$. By setting up a system of equations with the Pythagorean theorem, calculate the height of the trapezoid.
Two towers $15\text{ m}$ and $20\text{ m}$ high are separated by an unknown horizontal distance. A bird flies in a straight line from the top of the $15\text{ m}$ tower…
Two towers $15\text{ m}$ and $20\text{ m}$ high are separated by an unknown horizontal distance. A bird flies in a straight line from the top of the $15\text{ m}$ tower to the base of the $20\text{ m}$ tower, covering exactly $25\text{ m}$. Calculate the distance between the bases of the towers. Next, if another bird flies in a straight line from the top of the $20\text{ m}$ tower to the base of the $15\text{ m}$ tower, what distance will it cover?