# 63 Homework And Practice The Pythagorean Theorem

## 63 Homework and Practice The Pythagorean Theorem

The Pythagorean theorem is a fundamental relation in Euclidean geometry that relates the lengths of the sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The theorem can be written as an equation: a + b = c, where a and b are the lengths of the legs and c is the length of the hypotenuse.

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The Pythagorean theorem is named after the Greek philosopher and mathematician Pythagoras, who is credited with its proof, although it was known to other ancient civilizations, such as the Babylonians, Egyptians, and Indians. There are many different ways to prove the Pythagorean theorem, using geometry, algebra, or calculus. Some proofs are simple and elegant, while others are more complex and creative. One famous proof is attributed to US President James Garfield, who used a trapezoid to show that the areas of two squares are equal to the area of a larger square. Another proof is by the Indian mathematician Bhaskara, who used a clever rearrangement of four identical right triangles to form two squares with equal areas. You can find more proofs of the Pythagorean theorem on [Wikipedia] or [Khan Academy].

The Pythagorean theorem has many applications in mathematics and science. It can be used to find the distance between two points in a coordinate plane, using the formula: d = sqrt((x2 - x1) + (y2 - y1)), where (x1, y1) and (x2, y2) are the coordinates of the points. It can also be used to find the length of a missing side of a right triangle, given the lengths of the other two sides. For example, if a right triangle has legs of length 3 cm and 4 cm, then we can use the Pythagorean theorem to find that the hypotenuse has length 5 cm: c = 3 + 4, so c = sqrt(9 + 16), which is c = sqrt(25), or c = 5. Similarly, if we know that a right triangle has a hypotenuse of length 13 cm and one leg of length 5 cm, then we can use the Pythagorean theorem to find that the other leg has length 12 cm: b = 13 - 5, so b = sqrt(169 - 25), which is b = sqrt(144), or b = 12.

The Pythagorean theorem can also be extended to three-dimensional space, where it relates the lengths of the edges of a right rectangular prism. It states that the square of the diagonal (the longest edge) is equal to the sum of the squares of the other three edges. The formula is: d = l + w + h, where d is the length of the diagonal, and l, w, and h are the lengths of the length, width, and height respectively. For example, if a right rectangular prism has dimensions of 6 cm by 8 cm by 10 cm, then we can use the Pythagorean theorem to find that its diagonal has length 14.14 cm: d= 6+ 8+ 10, so d = sqrt(36 + 64 + 100), which is d = sqrt(200), or d = 14.14.

As you can see, the Pythagorean theorem is a very useful and powerful tool in mathematics and science. It can help you solve problems involving right triangles, distances, and three-dimensional shapes. To practice using the Pythagorean theorem, you can try some of the exercises on [BYJU'S] or [Math is Fun]. You can also explore some of the interactive simulations on [GeoGebra] or [PhET] to see how the Pythagorean theorem works visually. Have fun learning and applying the Pythagorean theorem!

References:

: Pythagorean theorem - Wikipedia

: Pythagorean theorem proofs - Khan Academy

: Pythagoras Theorem - BYJU'S

: Pythagoras' Theorem - Math is Fun

: Pythagorean Theorem - GeoGebra

: Area Model: Algebra - PhET