The definition of the Pythagorean theorem, or Pythagoras' theorem, states that the sum of the squares of the two legs of a right triangle is equal to the sum of the square of the hypotenuse. The hypotenuse is the longest side of a right triangle and is always opposite the right angle. This theorem is used in determining the length of the sides of a right triangle. The formula for the Pythagorean theorem is {eq}a^2 + b^2 = c^2 {/eq}, where a and b are the legs and c is the hypotenuse.
Pythagoras was a fifth century BCE Greek philosopher and mathematician who is credited with several scientific and mathematical discoveries, most notably the Pythagorean theorem. Although the Pythagorean theorem was popularized by Pythagoras, other ancient mathematicians in ancient civilizations such as Babylon and China had previously formulated the idea. Pythagoras was the first to prove its validity for all right triangles.
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Try it nowThe Pythagorean theorem can be proven in several different ways. To prove the theorem algebraically, begin with a diagram of a tilted square inside another square.
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By tilting the inner square so that its corners touch the sides of the outer square, four congruent right triangles are formed, each with sides a and b and a hypotenuse c. The sides of the larger square are equal to a + b, and the sides of the inner square are equal to c. The area of the larger square can be determined in two ways: {eq}A = (a + b)^2 {/eq} OR the sum of the areas of the four triangles and the inner square. Since both methods produce the same result, the two methods are equal. So, mathematically,
{eq}(a + b)^2 = 4(ab/2) + c^2\\\ a^2 + 2ab + b^2 = 2ab + c^2\\\ a^2 + b^2 = c^2 {/eq}
Another way to prove Pythagoras' theorem is by using the similar triangle method. Start by constructing a right triangle. Then, from the vertex of the right angle, draw a line perpendicular to the hypotenuse, as indicated in the diagram.
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For triangles ABC and ADB:
| angle A = angle A | common angle |
| angle ADB = angle ABC | right angles |
| triangle ADB is similar to triangle ABC | angle-angle similarity |
For triangles BDC and ABC:
| angle C = angle C | common angle |
| angle CDB = angle CBA | right angles |
| triangle BDC is similar to triangle ABC | angle-angle similarity |
Thus, all three triangles are similar, so their sides are proportionate.
| AB/AC = AD/AB, so AB2 = AD ⋅ AC | cross multiplication |
| BC/AC = CD/BC, so BC2 = CD ⋅ AC | cross multiplication |
| AB2 + BC2 = (AD ⋅ AC) + (CD ⋅ AC) | add both equations together |
| AB2 + BC2 = AC(AD + CD) | factoring |
| AB2 + BC2 = AC(AC) | substitution |
| AB2 + BC2 = AC2 | multiplication |
Pythagoras' theorem is proven!
Another mathematician, Euclid, proved the theorem in yet another way by constructing squares from each of the sides of the right triangle. Euclid's method is often referred to as the Windmill proof because its shape resembles that of a windmill.
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Since the Pythagorean theorem has been proven valid by many different methods, the formula {eq}a^2 + b^2 = c^2 {/eq} can be reliably used to find the missing side length of a right triangle. The length of a hypotenuse can be calculated using the formula when the other two side lengths are given, or the length of one leg can be calculated if only a leg length and hypotenuse are given. For example, use Pythagoras' formula to find the hypotenuse when the lengths of the other two sides are 3 and 4. To find the hypotenuse, substitute 3 and 4 for a and b, then solve for c.
{eq}3^2 + 4^2 = c^2\\\ 9 + 16 = c^2\\\ 25 = c^2\\\ 5 = c {/eq}
Note: When practicing how to do the Pythagorean theorem, be sure to square the a and b terms before adding them together.
Some geometry problems require determining which type of triangle is present. For these types of problems, the converse of Pythagoras' theorem will be helpful. The converse of the Pythagorean theorem states that if the sum of the squares of the shorter two legs of a triangle equal the square of the longer side, then the triangle is a right triangle.
A Pythagorean triple consists of three positive integers that satisfy the Pythagorean formula. The smallest Pythagorean triple is 3-4-5 as {eq}3^2 + 4^2 = 5^2 {/eq}, and another example is 5-12-13 because {eq}5^2 + 12^2 = 13^2 {/eq}. A Pythagorean square is a tool that may be helpful when using the Pythagorean theorem. The Pythagorean square is a multiplication table in the shape of a square, where the perfect squares are on the diagonal from top left to bottom right. This tool may be helpful in quickly determining the squares of numbers.
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To reinforce how the Pythagorean theorem works, follow along with these examples:
Example 1: Find the length of side a of a triangle with side b = 7 and a hypotenuse of 10.
Solution: First plug the numbers into the formula. Then, simplify and solve for the variable a.
{eq}a^2 + 7^2 = 10^2\\\ a^2 + 49 = 100\\\ a^2 = 51\\\ a = 7.14 {/eq}
Example 2: Find the hypotenuse of an isosceles right triangle with leg lengths of 13.
Solution: The legs of an isosceles right triangle are both the same, so
{eq}13^2 + 13^2 = c^2\\\ 169 +169 = c^2\\\ 338 = c^2\\\ 18.38 = c {/eq}
Example 3: If the length of the diagonal of a rectangle is 12 and the rectangle's width is 8, what is the length (b) of the rectangle?
Solution: Because a rectangle has four right angles, the diagonal of the rectangle is the same as the hypotenuse of the two triangles it forms. So, the length of the rectangle can be found using the Pythagorean theorem.
{eq}8^2 + b^2 = 12^2\\\ 64 + b^2 = 144\\\ b^2 = 80\\\ b = 8.94 {/eq}
In addition to being a useful tool in solving geometry problems, the Pythagorean theorem is useful in many every day and professional settings. Construction workers and painters may use the theorem to figure out how long of a ladder is needed to reach certain heights. Engineers rely on the theorem in designing bridges, roads, and maps. The theorem is also helpful in navigation, assisting in determining the shortest distance or most efficient path.
The Pythagorean theorem, first proven by Greek philosopher and mathematician Pythagoras, states that for all right triangles, the sum of the squares of the shorter two sides is equal to the square of the hypotenuse. The validity of the theorem has been proven in a number of different ways by several mathematicians after Pythagoras, including Euclid. Pythagoras' theorem is helpful not only within the confines of the study of geometry, but also in every day life and in many professions.
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To find c, or the hypotenuse, first find the squares of a and b. Then, find the square root of the sum of those squares.
Pythagoras' theorem is used for finding the length of a missing side of a right triangle. It can be used to find a leg length or the hypotenuse length.
The formula for the Pythagorean theorem is the sum of the squares of the two legs equal to the square of the hypotenuse. Mathematically, the formula is {eq}a^2 + b^2 = c^2 {/eq}, where a and b are the legs and c is the hypotenuse.
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