Pythagorean Theorem

Pythagorean Theorem - Statement and Proof

The Pythagorean Theorem is undoubtedly one of the most famous and fundamental theorems in all of mathematics, particularly in geometry. It reveals a crucial relationship that exists exclusively in right-angled triangles. Named after the ancient Greek mathematician Pythagoras, who is traditionally credited with its first proof (though the relationship was known in other ancient civilizations like Babylon and India centuries earlier), this theorem is indispensable for solving problems involving lengths in right triangles.

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Theorem Statement

The Pythagorean theorem states the following property for any right-angled triangle:

Theorem: In a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (which are called the legs or catheti).

Consider a triangle $\triangle \text{ABC}$ that is right-angled at vertex B. Let the lengths of the sides opposite to vertices A, B, and C be denoted by the corresponding lowercase letters $a, b$, and $c$ respectively. In this case, the side opposite the right angle (at B) is AC, which is the hypotenuse. The sides opposite the other two angles, AB and BC, are the legs.

The theorem can be expressed as an equation:

Using the standard notation for $\triangle \text{ABC}$ right-angled at B:

Or, using side lengths $b$ (hypotenuse opposite B), $c$ (opposite C), and $a$ (opposite A) where the right angle is at A:

Let's stick to the notation where the right angle is at B, and the sides are AB, BC, and AC.

In this figure, $\triangle \text{ABC}$ is right-angled at B. The side opposite the right angle is AC (hypotenuse). The sides AB and BC are the legs. The theorem states $\text{AC}^2 = \text{AB}^2 + \text{BC}^2$. If we label the lengths $AC=c$, $BC=a$, $AB=b$, then the theorem is $c^2 = a^2 + b^2$. However, the image provided labels the sides opposite vertices A, B, C as a, b, c respectively. If the right angle is at B, then the side opposite B is b (AC, the hypotenuse), opposite A is a (BC, a leg), and opposite C is c (AB, a leg). So, for the given image notation, the theorem is $b^2 = a^2 + c^2$. Let's clarify and use the notation from the proof description later (AB, BC, AC as side names).

Let's re-state using vertex names:

In $\triangle \text{ABC}$, if $\angle \text{B}$ is the right angle ($90^\circ$), then $(\text{length of AC})^2 = (\text{length of AB})^2 + (\text{length of BC})^2$.

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Proof using Similar Triangles

There are numerous proofs of the Pythagorean theorem. One elegant method involves using the properties of similar triangles.

Given: A right-angled triangle $\triangle \text{ABC}$, right-angled at vertex B.

To Prove: $\text{AC}^2 = \text{AB}^2 + \text{BC}^2$

Construction: Draw a perpendicular line segment from the vertex B to the hypotenuse AC. Let this perpendicular intersect AC at point D.

This construction creates two smaller right-angled triangles, $\triangle \text{ADB}$ and $\triangle \text{BDC}$, and the original triangle $\triangle \text{ABC}$. All three of these triangles are similar to each other.

Proof:

Consider $\triangle \text{ADB}$ and $\triangle \text{ABC}$.

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Now, consider $\triangle \text{BDC}$ and $\triangle \text{ABC}$.

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Thus, the square of the hypotenuse (AC) is equal to the sum of the squares of the other two sides (AB and BC).

Hence Proved.

Pythagorean Theorem - Converse of the Pythagorean Theorem

The Pythagorean Theorem provides a property that is true for all right-angled triangles. The converse of this theorem is equally important and useful. It allows us to determine if a triangle is a right-angled triangle just by knowing the lengths of its three sides. This is a powerful tool for classifying triangles based on their side lengths.

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Statement of the Converse of the Pythagorean Theorem

The converse of the Pythagorean theorem states:

Theorem: If in a triangle, the square of the length of one side is equal to the sum of the squares of the lengths of the other two sides, then the angle opposite the first side is a right angle ($90^\circ$), and the triangle is a right-angled triangle.

Consider a triangle $\triangle \text{ABC}$ with side lengths $a, b,$ and $c$, where $c$ is the length of the longest side. If you find that the relationship $c^2 = a^2 + b^2$ holds true for these side lengths, then the angle opposite the side with length $c$ (which is angle C, $\angle \text{C}$) must be a right angle ($90^\circ$).

Alternatively, using vertex names for the sides:

If in $\triangle \text{ABC}$, the square of the length of side AC is equal to the sum of the squares of the lengths of sides AB and BC (i.e., $\text{AC}^2 = \text{AB}^2 + \text{BC}^2$), then the angle opposite side AC, which is $\angle \text{B}$, is a right angle ($90^\circ$).

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Proof of the Converse of the Pythagorean Theorem

We can prove the converse of the Pythagorean theorem by constructing a right-angled triangle with the given side lengths and showing that it is congruent to the original triangle.

Given: A triangle $\triangle \text{ABC}$ such that the square of the length of side AC is equal to the sum of the squares of the lengths of sides AB and BC ($\text{AC}^2 = \text{AB}^2 + \text{BC}^2$).

To Prove: $\triangle \text{ABC}$ is a right-angled triangle, specifically right-angled at vertex B (i.e., $\text{m}\angle \text{B} = 90^\circ$).

Construction: Construct a new triangle $\triangle \text{PQR}$ that is right-angled at vertex Q. Ensure that the lengths of the legs of $\triangle \text{PQR}$ are equal to the lengths of two sides of $\triangle \text{ABC}$, specifically construct it such that $\text{PQ} = \text{AB}$ and $\text{QR} = \text{BC}$.

Proof:

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This proves that the angle at vertex B in $\triangle \text{ABC}$ is a right angle ($90^\circ$). Consequently, $\triangle \text{ABC}$ is a right-angled triangle.

Hence Proved.

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Application of the Converse

The converse of the Pythagorean theorem is widely used to check if a triangle is right-angled, given its side lengths. For example, a triangle with side lengths 3, 4, and 5 units is a right-angled triangle because $3^2 + 4^2 = 9 + 16 = 25 = 5^2$. The angle opposite the side of length 5 is the right angle.

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Pythagorean Theorem - Applications of the Pythagorean Theorem

The Pythagorean Theorem is not just a fundamental result in geometry; it is also one of the most widely applicable mathematical theorems across numerous fields. Its power lies in providing a direct relationship between lengths in right-angled triangles, which appear frequently in various contexts, whether physical or abstract.

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Calculating Side Lengths in Right-Angled Triangles

The most immediate application of the Pythagorean theorem is to find the length of an unknown side of a right-angled triangle when the lengths of the other two sides are known. Let $a$ and $b$ be the lengths of the legs, and $c$ be the length of the hypotenuse in a right-angled triangle.

• Finding the Hypotenuse: If the lengths of the two legs, $a$ and $b$, are known, the length of the hypotenuse, $c$, can be found using the formula derived from $a^2 + b^2 = c^2$:

  $c = \sqrt{a^2 + b^2}$

• Finding a Leg: If the length of the hypotenuse, $c$, and the length of one leg (say, $a$) are known, the length of the other leg, $b$, can be found by rearranging the formula $a^2 + b^2 = c^2$: $b^2 = c^2 - a^2$. Taking the square root gives:

  $b = \sqrt{c^2 - a^2}$

  Similarly, if leg $b$ is known, leg $a = \sqrt{c^2 - b^2}$.

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Coordinate Geometry (Distance Formula)

One of the most significant applications of the Pythagorean theorem is in establishing the distance formula in a Cartesian coordinate system (analytic geometry). The distance between any two points in a 2D plane can be found by considering them as vertices of a right-angled triangle.

Let $P_1 = (x_1, y_1)$ and $P_2 = (x_2, y_2)$ be two points in the Cartesian plane. We can form a right-angled triangle by drawing a horizontal line through one point and a vertical line through the other, meeting at a third point, say $P_3 = (x_2, y_1)$. The horizontal leg of this triangle has length $|x_2 - x_1|$, and the vertical leg has length $|y_2 - y_1|$. The distance between $P_1$ and $P_2$ is the hypotenuse of this right triangle.

Applying the Pythagorean theorem, if $d$ is the distance between $(x_1, y_1)$ and $(x_2, y_2)$:

Taking the square root gives the distance formula:

This formula is widely used in coordinate geometry and other areas of mathematics and science.

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Architecture and Construction

The Pythagorean theorem is invaluable in architecture and construction for ensuring square corners and checking the alignment of structural elements. Builders use it frequently, especially the converse theorem and Pythagorean triples.

For example, to verify that a wall is exactly perpendicular to the floor (forms a right angle), a builder can measure a distance up the wall (say 4 units, like 4 feet or 40 cm) and a distance out along the floor from the base of the wall (say 3 units). If the diagonal distance measured between these two points is exactly 5 units, then the corner is indeed a right angle ($3^2 + 4^2 = 9 + 16 = 25 = 5^2$). This simple check using the (3, 4, 5) Pythagorean triple ensures structural integrity where right angles are required.

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Navigation

The theorem is used in navigation to calculate the shortest distance between two points or to determine a displacement. If a ship or aircraft travels a certain distance east and then a certain distance north, the direct distance from the starting point to the end point is the hypotenuse of a right triangle formed by the eastward and northward legs.

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Applications in Various Geometry Problems

• Finding the Diagonal of a Rectangle or Square: The diagonal of a rectangle forms the hypotenuse of a right triangle with the two adjacent sides as legs. For a rectangle with length $l$ and width $w$, the diagonal $d = \sqrt{l^2 + w^2}$. For a square with side $s$, $d = \sqrt{s^2 + s^2} = \sqrt{2s^2} = s\sqrt{2}$.
• Finding the Height of Triangles: The height of an isosceles or equilateral triangle often forms a right-angled triangle with half of the base and one of the sides. The Pythagorean theorem can be used to calculate this height.
• Calculating Lengths in 3D Geometry: The theorem can be extended to find distances in three-dimensional space. For example, the space diagonal (the diagonal connecting opposite vertices passing through the interior) of a rectangular prism (cuboid) with dimensions $l, w,$ and $h$ is found by applying the Pythagorean theorem twice: first to find the diagonal of the base ($\sqrt{l^2 + w^2}$), and then using this result and the height $h$ as legs of another right triangle. The space diagonal $D = \sqrt{(\sqrt{l^2 + w^2})^2 + h^2} = \sqrt{l^2 + w^2 + h^2}$.

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Pythagorean Triples

A set of three positive integers, traditionally denoted as $(a, b, c)$, is called a Pythagorean triple if they satisfy the equation $a^2 + b^2 = c^2$. These triples represent the side lengths of right-angled triangles where all sides have integer lengths. Pythagorean triples are useful in problems and constructions as they simplify calculations.

If $(a, b, c)$ is a Pythagorean triple, then $(ka, kb, kc)$ is also a Pythagorean triple for any positive integer $k$. This means any multiple of a Pythagorean triple also forms the sides of a right-angled triangle. A triple is considered primitive if the three integers $a, b,$ and $c$ have no common divisor greater than 1.

Some common primitive Pythagorean triples include:

• (3, 4, 5) $\implies 3^2 + 4^2 = 9 + 16 = 25 = 5^2$
• (5, 12, 13) $\implies 5^2 + 12^2 = 25 + 144 = 169 = 13^2$
• (8, 15, 17) $\implies 8^2 + 15^2 = 64 + 225 = 289 = 17^2$
• (7, 24, 25) $\implies 7^2 + 24^2 = 49 + 576 = 625 = 25^2$
• (20, 21, 29) $\implies 20^2 + 21^2 = 400 + 441 = 841 = 29^2$

Recognizing these triples can sometimes provide a shortcut in solving problems involving right triangles.

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