Step 3 PU² = 81+ 199 * = √223 4 Step 3 N = 289-22 4 Step 4 U = √√225- V223 Step 4 N V64 17 2. N² + 15 Step 5 X = 15 Step 5 N = 8 ,, By Using The Pythagorean Theorem, Look For The Lost Side Length Of

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Introduction

The Pythagorean theorem is a fundamental concept in geometry that has been used for centuries to calculate the length of the sides of right-angled triangles. The theorem states that 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. In mathematical terms, this can be expressed as a² + b² = c², where a and b are the lengths of the two sides and c is the length of the hypotenuse. In this article, we will explore how to use the Pythagorean theorem to solve for lost side lengths, and provide a step-by-step guide to help you master this essential mathematical concept.

Step 3: PU² = 81 + 199 * = √223

To begin, let's examine the given equation: PU² = 81 + 199 * = √223. This equation represents a right-angled triangle with one side length missing. Our goal is to solve for the missing side length, denoted by the variable P.

PU² = 81 + 199 * = √223

To solve for P, we can start by isolating the variable P on one side of the equation. We can do this by subtracting 81 from both sides of the equation, which gives us:

PU² - 81 = 199 * = √223

Next, we can simplify the left-hand side of the equation by factoring out the variable P:

P²U² - 81 = 199 * = √223

Now, we can see that the left-hand side of the equation is a quadratic expression in terms of P. To solve for P, we can use the quadratic formula:

P = (-b ± √(b² - 4ac)) / 2a

In this case, the coefficients of the quadratic expression are a = U², b = 0, and c = -81. Plugging these values into the quadratic formula, we get:

P = (0 ± √(0² - 4U²(-81))) / 2U²

Simplifying the expression under the square root, we get:

P = (0 ± √(324U²)) / 2U²

Now, we can simplify the expression further by factoring out the square root:

P = (0 ± 18U) / 2U²

Since we are looking for the positive value of P, we can ignore the negative solution and simplify the expression to:

P = 18U / 2U²

Finally, we can simplify the expression by canceling out the common factor of U:

P = 9 / U

Step 3: N = 289 - 22

Now, let's move on to the next step, which involves solving for the variable N. The equation given is N = 289 - 22.

N = 289 - 22

To solve for N, we can simply subtract 22 from 289:

N = 267

Step 4: U = √√225 - V223

Next, we have the equation U = √√225 - V223. This equation represents a right-angled triangle with one side length missing. Our goal is to solve for the missing side length, denoted by the variable U.

U = √√225 - V223

To solve for U, we can start by simplifying the expression under the square root:

U = √(√225) - V223

Next, we can simplify the expression further by evaluating the square root of 225:

U = √15 - V223

Now, we can see that the expression under the square root is a perfect square, which means that we can simplify it further:

U = 3 - V223

Step 4: NV64 = 17

Next, we have the equation NV64 = 17. This equation represents a right-angled triangle with one side length missing. Our goal is to solve for the missing side length, denoted by the variable N.

NV64 = 17

To solve for N, we can start by isolating the variable N on one side of the equation. We can do this by dividing both sides of the equation by 64:

N = 17 / 64

Now, we can simplify the expression by evaluating the division:

N = 0.265625

Step 5: x = 15

Next, we have the equation x = 15. This equation represents a right-angled triangle with one side length missing. Our goal is to solve for the missing side length, denoted by the variable x.

x = 15

To solve for x, we can simply assign the value 15 to the variable x:

x = 15

Step 5: N = 8

Finally, we have the equation N = 8. This equation represents a right-angled triangle with one side length missing. Our goal is to solve for the missing side length, denoted by the variable N.

N = 8

To solve for N, we can simply assign the value 8 to the variable N:

N = 8

Conclusion

Q: What is the Pythagorean theorem?

A: The Pythagorean theorem is a fundamental concept in geometry that describes the relationship between the lengths of the sides of a right-angled triangle. It states that 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.

Q: How do I use the Pythagorean theorem to solve for lost side lengths?

A: To use the Pythagorean theorem to solve for lost side lengths, you need to follow these steps:

  1. Identify the lengths of the two sides of the right-angled triangle.
  2. Use the Pythagorean theorem to calculate the length of the hypotenuse.
  3. Simplify the expression by canceling out common factors.
  4. Solve for the missing side length.

Q: What are some common mistakes to avoid when using the Pythagorean theorem?

A: Some common mistakes to avoid when using the Pythagorean theorem include:

  • Not simplifying the expression by canceling out common factors.
  • Not using the correct formula (a² + b² = c²).
  • Not identifying the lengths of the two sides of the right-angled triangle.
  • Not solving for the missing side length.

Q: Can I use the Pythagorean theorem to solve for non-right-angled triangles?

A: No, the Pythagorean theorem can only be used to solve for right-angled triangles. If you are dealing with a non-right-angled triangle, you will need to use a different method to solve for the side lengths.

Q: How do I know if a triangle is a right-angled triangle?

A: To determine if a triangle is a right-angled triangle, you can use the following criteria:

  • The triangle has one right angle (90 degrees).
  • The sum of the squares of the lengths of the two sides is equal to the square of the length of the hypotenuse (a² + b² = c²).

Q: Can I use the Pythagorean theorem to solve for the length of the hypotenuse?

A: Yes, you can use the Pythagorean theorem to solve for the length of the hypotenuse. Simply rearrange the formula to solve for c (c = √(a² + b²)).

Q: What are some real-world applications of the Pythagorean theorem?

A: The Pythagorean theorem has many real-world applications, including:

  • Building design and construction.
  • Engineering and architecture.
  • Physics and mathematics.
  • Navigation and surveying.

Q: Can I use the Pythagorean theorem to solve for complex numbers?

A: Yes, you can use the Pythagorean theorem to solve for complex numbers. However, you will need to use a different formula (a² + b² = c², where a and b are complex numbers).

Q: How do I simplify complex expressions using the Pythagorean theorem?

A: To simplify complex expressions using the Pythagorean theorem, you can use the following steps:

  1. Identify the complex numbers in the expression.
  2. Use the Pythagorean theorem to calculate the length of the hypotenuse.
  3. Simplify the expression by canceling out common factors.
  4. Solve for the missing side length.

Conclusion

In this article, we have answered some of the most frequently asked questions about the Pythagorean theorem. We have covered topics such as how to use the Pythagorean theorem to solve for lost side lengths, common mistakes to avoid, and real-world applications of the theorem. By following the steps outlined in this article, you should be able to use the Pythagorean theorem with confidence and accuracy.