Problem 23 In your own words, explain the P... [FREE SOLUTION]

Chapter 7: Problem 23

In your own words, explain the Pythagorean theorem.

Short Answer

Expert verified

The Pythagorean theorem states that in any right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). This can be represented as \(c^2 = a^2 + b^2\). It can be used to solve geometrical problems involving right-angled triangles, such as finding the length of an unknown side given the lengths of the other two sides. For example, in a right-angled triangle with side lengths a = 3, b = 4, and c = 5, the theorem can be verified as \(25 = 9 + 16\).

Step by step solution

Understand the Pythagorean theorem

The Pythagorean theorem, named after the ancient Greek mathematician Pythagoras, states that in any 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 (the legs). The theorem can be written as an equation relating the lengths of the sides a, b, and c, often called the Pythagorean equation: \[c^2 = a^2 + b^2\] Here, c is the length of the hypotenuse, and a and b are the lengths of the other two sides.

Example triangle

Let's consider a right-angled triangle with side lengths a = 3, b = 4, and c = 5.

Verify the Pythagorean equation

To verify that the given triangle adheres to the Pythagorean theorem, we need to check if the sum of the squares of the lengths of sides a and b equals the square of the length of side c. This can be done using the following calculation: \[c^2 = a^2 + b^2\] \[5^2 = 3^2 + 4^2\] \[25 = 9 + 16\] \[25 = 25\] The equation holds true for the given triangle, which confirms that the Pythagorean theorem applies to this case. This test can be performed on any right-angled triangle to verify the theorem.

Using the Pythagorean theorem in problem-solving

The Pythagorean theorem can be used to solve various geometrical problems involving right-angled triangles. For instance, if we are given the lengths of two sides of a right-angled triangle and are asked to find the length of the third side, we can simply plug the given values into the Pythagorean equation and solve for the unknown side length: For example, consider a right-angled triangle with side lengths a = 6 and b = 8. To find the length of the hypotenuse, c, we can use the Pythagorean equation: \[c^2 = a^2 + b^2\] \[c^2 = 6^2 + 8^2\] \[c^2 = 36 + 64\] \[c^2 = 100\] \[c = \sqrt{100}\] \[c = 10\] Thus, the length of the hypotenuse in this right-angled triangle is 10.

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In your own words, explain the Pythagorean theorem.

Short Answer

Step by step solution

Understand the Pythagorean theorem

Example triangle

Verify the Pythagorean equation

Using the Pythagorean theorem in problem-solving

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