Derivation of the Pythagorean Theorem

Oct 13, 2025 by miodeya.com 92 views

Introduction

The Pythagorean theorem is a fundamental concept in mathematics that helps us find the length of the hypotenuse of a right-angled triangle. The theorem states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. In this article, we will analyze a given solution to a problem involving the Pythagorean theorem and identify the errors in the solution. We will also provide corrections to the solution.

The Given Solution

The given solution is as follows:

\begin{aligned} 8^2 + 12^2 & = d^2 \\ 64 + 144 & = d^2 \\ 208 & = d^2 \\ \sqrt{208} & = d \\ 14.42 & \approx d \end{aligned}

Error 1: Lack of Calculation for the Other Side

The first error in the given solution is that it only calculates the length of the hypotenuse (d) and does not find the full distance each side. To find the full distance each side, we need to calculate the length of the other two sides using the Pythagorean theorem.

Correction 1: Calculating the Length of the Other Two Sides

Let's assume that the length of the other two sides are x and y. Using the Pythagorean theorem, we can write:

x^2 + y^2 = d^2

We are given that $d^2 = 208$ . Therefore, we can write:

x^2 + y^2 = 208

To find the length of the other two sides, we need to find the values of x and y that satisfy the equation.

Error 2: Approximation of the Square Root

The second error in the given solution is that it approximates the square root of 208 to 14.42. However, the square root of 208 is not exactly equal to 14.42. A more accurate approximation of the square root of 208 is 14.42 (rounded to two decimal places).

Correction 2: Finding the Exact Value of the Square Root

To find the exact value of the square root of 208, we can use a calculator or a mathematical software. The exact value of the square root of 208 is:

\sqrt{208} = \sqrt{16 \times 13} = 4\sqrt{13}

Error 3: Lack of Verification

The third error in the given solution is that it does not verify the result. To verify the result, we need to check if the calculated value of d satisfies the Pythagorean theorem.

Correction 3: Verification of the Result

To verify the result, we need to check if the calculated value of d satisfies the Pythagorean theorem. We can do this by plugging the value of d into the Pythagorean theorem and checking if the equation holds true.

Conclusion

In conclusion, the given solution to the problem involving the Pythagorean theorem contains three errors: lack of calculation for the other side, approximation of the square root, and lack of verification. We have provided corrections to the solution and highlighted the importance of verifying the result.

Pythagorean Theorem: Formula and Derivation

a^2 + b^2 = c^2

where a and b are the lengths of the other two sides, and c is the length of the hypotenuse.

Derivation of the Pythagorean Theorem

The Pythagorean theorem can be derived using the concept of similar triangles. Let's consider a right-angled triangle with legs of length a and b, and a hypotenuse of length c. We can draw a line from the vertex of the right angle to the hypotenuse, which divides the triangle into two smaller triangles. These smaller triangles are similar to each other and to the original triangle.

Using the concept of similar triangles, we can write:

\frac{a}{b} = \frac{b}{c}

Squaring both sides of the equation, we get:

\frac{a^2}{b^2} = \frac{b^2}{c^2}

Cross-multiplying, we get:

a^2c^2 = b^2b^2

Simplifying, we get:

a^2 + b^2 = c^2

This is the Pythagorean theorem.

Applications of the Pythagorean Theorem

The Pythagorean theorem has numerous applications in mathematics and science. Some of the applications of the Pythagorean theorem include:

Geometry: The Pythagorean theorem is used to find the length of the hypotenuse of a right-angled triangle.
Trigonometry: The Pythagorean theorem is used to find the values of trigonometric functions such as sine, cosine, and tangent.
Physics: The Pythagorean theorem is used to find the distance and displacement of objects in motion.
Engineering: The Pythagorean theorem is used to design and build structures such as bridges, buildings, and roads.

Conclusion

Q: What is the Pythagorean theorem?

A: The Pythagorean theorem is a fundamental concept in mathematics that helps us find the length of the hypotenuse of a right-angled triangle. The theorem states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Q: How do I use the Pythagorean theorem?

A: To use the Pythagorean theorem, you need to know the lengths of the other two sides of the right-angled triangle. You can then plug these values into the formula:

a^2 + b^2 = c^2

where a and b are the lengths of the other two sides, and c is the length of the hypotenuse.

Q: What is the formula for the Pythagorean theorem?

A: The formula for the Pythagorean theorem is:

a^2 + b^2 = c^2

where a and b are the lengths of the other two sides, and c is the length of the hypotenuse.

Q: Can I use the Pythagorean theorem to find the length of a side of a triangle that is not a right-angled triangle?

A: No, the Pythagorean theorem can only be used to find the length of the hypotenuse of a right-angled triangle. If you have a triangle that is not a right-angled triangle, you will need to use a different method to find the length of the sides.

Q: How do I find the length of the hypotenuse of a right-angled triangle if I only know the lengths of the other two sides?

A: To find the length of the hypotenuse of a right-angled triangle if you only know the lengths of the other two sides, you can use the Pythagorean theorem. Simply plug the values of the other two sides into the formula:

a^2 + b^2 = c^2

where a and b are the lengths of the other two sides, and c is the length of the hypotenuse.

Q: Can I use the Pythagorean theorem to find the length of a side of a triangle if I only know the length of the hypotenuse?

A: No, the Pythagorean theorem can only be used to find the length of the hypotenuse of a right-angled triangle if you know the lengths of the other two sides. If you only know the length of the hypotenuse, you will need to use a different method to find the length of the sides.

Q: How do I find the length of the sides of a right-angled triangle if I only know the length of the hypotenuse?

A: To find the length of the sides of a right-angled triangle if you only know the length of the hypotenuse, you can use the fact that the sum of the squares of the lengths of the other two sides is equal to the square of the length of the hypotenuse. This can be expressed as:

a^2 + b^2 = c^2

where a and b are the lengths of the other two sides, and c is the length of the hypotenuse.

Q: Can I use the Pythagorean theorem to find the length of a side of a triangle if I know the lengths of two sides and the angle between them?

A: No, the Pythagorean theorem can only be used to find the length of the hypotenuse of a right-angled triangle if you know the lengths of the other two sides. If you know the lengths of two sides and the angle between them, you will need to use a different method to find the length of the sides.

Q: How do I find the length of the sides of a right-angled triangle if I know the lengths of two sides and the angle between them?

A: To find the length of the sides of a right-angled triangle if you know the lengths of two sides and the angle between them, you can use the fact that the sum of the squares of the lengths of the other two sides is equal to the square of the length of the hypotenuse. This can be expressed as:

a^2 + b^2 = c^2

where a and b are the lengths of the other two sides, and c is the length of the hypotenuse.

Q: Can I use the Pythagorean theorem to find the length of a side of a triangle if I know the lengths of three sides?

A: No, the Pythagorean theorem can only be used to find the length of the hypotenuse of a right-angled triangle if you know the lengths of the other two sides. If you know the lengths of three sides, you will need to use a different method to find the length of the sides.

Q: How do I find the length of the sides of a right-angled triangle if I know the lengths of three sides?

A: To find the length of the sides of a right-angled triangle if you know the lengths of three sides, you can use the fact that the sum of the squares of the lengths of the other two sides is equal to the square of the length of the hypotenuse. This can be expressed as:

a^2 + b^2 = c^2

where a and b are the lengths of the other two sides, and c is the length of the hypotenuse.

Conclusion

In conclusion, the Pythagorean theorem is a fundamental concept in mathematics that helps us find the length of the hypotenuse of a right-angled triangle. The theorem has numerous applications in mathematics and science, and is used to find the values of trigonometric functions, distance and displacement of objects in motion, and design and build structures.