Rectangle Problems: Find Truth Values

Hey guys! Let's dive into a fun geometry problem! We've got a rectangle here, and we're going to figure out some truths about it. The key is understanding how the properties of a rectangle work and using those to calculate the answer. We'll be using some basic math, so it should be a nice challenge.

We're given a rectangle with a length of $(8 + 4 \sqrt{2})$ cm and a diagonal of $(4\sqrt{5} + 2\sqrt{10})$ cm. Our mission? To determine whether several statements about this rectangle are true or false. Ready to get started? Let's go!

Understanding Rectangles and Key Concepts

Alright, before we jump into the statements, let's refresh our memory about rectangles. A rectangle is a four-sided shape (a quadrilateral) where all four angles are right angles (90 degrees). The opposite sides of a rectangle are equal in length, and the diagonals (lines drawn from one corner to the opposite corner) are also equal in length.

• Key Properties to Remember:

  • Opposite sides are equal in length.
  • All angles are 90 degrees.
  • Diagonals are equal in length.

• The Pythagorean Theorem: This is our best friend when dealing with right triangles, which is exactly what we get when we split a rectangle along its diagonal. The theorem states: $a^2 + b^2 = c^2$, where 'a' and 'b' are the lengths of the two shorter sides (legs), and 'c' is the length of the longest side (hypotenuse, which is the diagonal in our case).

• How We'll Use This Information: We know the length and the diagonal. We can use the Pythagorean Theorem to find the width. Once we have the length and width, we can check the statements given to us and determine if they're true or false. So it will be a bit of a process, but a great way to put those geometry and math skills to the test!

So, as a refresher, we have length $l = (8 + 4 \sqrt{2})$ cm and diagonal $d = (4 \sqrt{5} + 2\sqrt{10})$ cm. We need to find the width, w, and then we can evaluate our statements. Let's do this step by step, so we don't get lost in the numbers.

First, we need to find the width of the rectangle. To do that, we'll use the Pythagorean Theorem. We know that the length and width of a rectangle are the legs of a right triangle, and the diagonal is the hypotenuse. We can write the formula as: $l^2 + w^2 = d^2$. We can rearrange this to solve for the width: $w^2 = d^2 - l^2$, and $w = \sqrt{d^2 - l^2}$.

Now, let's plug in the values and calculate! This is where we need to be careful with our algebra and radicals, so we don't make any mistakes. Let's go through the detailed calculation step by step, so we don't miss any of the key values!

Calculating the Width of the Rectangle

Alright, let's get into the nitty-gritty and find that width! We've got our formula, $w = \sqrt{d^2 - l^2}$, and we know that $l = (8 + 4 \sqrt{2})$ cm and $d = (4 \sqrt{5} + 2 \sqrt{10})$ cm. Let's do the calculations and be extra careful to avoid any silly errors. Remember guys, it's all about precision in math!

1. Calculate $d^2$:

   $d^2 = (4 \sqrt{5} + 2 \sqrt{10})^2$

   $d^2 = (4 \sqrt{5})^2 + 2 * (4 \sqrt{5}) * (2 \sqrt{10}) + (2 \sqrt{10})^2$

   $d^2 = 16 * 5 + 16 \sqrt{50} + 4 * 10$

   $d^2 = 80 + 16 \sqrt{25 * 2} + 40$

   $d^2 = 120 + 16 * 5 \sqrt{2}$

   $d^2 = 120 + 80 \sqrt{2}$

2. Calculate $l^2$:

   $l^2 = (8 + 4 \sqrt{2})^2$

   $l^2 = 8^2 + 2 * 8 * 4 \sqrt{2} + (4 \sqrt{2})^2$

   $l^2 = 64 + 64 \sqrt{2} + 16 * 2$

   $l^2 = 64 + 64 \sqrt{2} + 32$

   $l^2 = 96 + 64 \sqrt{2}$

3. Calculate $w^2 = d^2 - l^2$:

   $w^2 = (120 + 80 \sqrt{2}) - (96 + 64 \sqrt{2})$

   $w^2 = 120 - 96 + 80 \sqrt{2} - 64 \sqrt{2}$

   $w^2 = 24 + 16 \sqrt{2}$

4. Calculate $w = \sqrt{w^2}$:

   $w = \sqrt{24 + 16 \sqrt{2}}$

   To simplify this, let's see if we can rewrite $24 + 16 \sqrt{2}$ as a square of a binomial, $(a + b)^2 = a^2 + 2ab + b^2$:

   We can guess and check, or we can look for two numbers that multiply to give us the same radical when squared. Let's try expressing $w$ in the form $(a + b \sqrt{2})$:

   $w = a + b \sqrt{2}$

   $w^2 = (a + b \sqrt{2})^2 = a^2 + 2ab \sqrt{2} + 2b^2$

   We want $a^2 + 2b^2 = 24$ and $2ab = 16$, which means $ab = 8$. From $ab = 8$, we can have a few possibilities (1, 8), (2, 4), (4, 2), or (8, 1). Let's check these pairs in $a^2 + 2b^2 = 24$. If $a = 4$ and $b = 2$, we have $4^2 + 2(2^2) = 16 + 8 = 24$, which is the correct match. So we have $a=4$ and $b=2$.

   Therefore, $w = 4 + 2 \sqrt{2}$

Now that we have the width! That was a little bit of work, but we got there. Now we can finally start checking those statements, and it should be much easier with the values calculated. Always remember to break down the problems into smaller, manageable steps. Remember to be patient and double-check your calculations, guys!

Evaluating the Statements

Alright, now that we have the width ($w = 4 + 2 \sqrt{2}$ cm), let's get down to business and figure out whether the statements are true or false. We will carefully analyze each one, using the values we've calculated, and see what the truth value is for each of the statements. Ready to roll?

Let's evaluate each statement!

• Statement 1: The length of the rectangle is 2 cm. We know the length of the rectangle is $(8 + 4 \sqrt{2})$ cm. Since $8 + 4 \sqrt{2}$ is not equal to 2, this statement is false. We can write 4 * 1.41 = 5.64, so the total length is 8 + 5.64 = 13.64 cm. So the statement is false. Guys, always make sure you know your values before declaring anything!

• Statement 2: The width of the rectangle is 4 cm. We calculated the width to be $4 + 2 \sqrt{2}$ cm. Therefore, the statement is false. $2 * \sqrt{2}$ is about $2 * 1.41 = 2.82$. So it is not 4 cm, therefore, the statement is false.

• Statement 3: The perimeter of the rectangle is 20 cm. The perimeter of a rectangle is calculated as $2 * (length + width)$. The length is $(8 + 4 \sqrt{2})$ cm and the width is $(4 + 2 \sqrt{2})$ cm. Let's calculate the perimeter:

  Perimeter = $2 * ((8 + 4 \sqrt{2}) + (4 + 2 \sqrt{2}))$

  Perimeter = $2 * (12 + 6 \sqrt{2})$

  Perimeter = $24 + 12 \sqrt{2}$

  Since $24 + 12 \sqrt{2}$ is not equal to 20, this statement is false. Let's calculate the real values. $12 * 1.41 = 16.92$, so $24 + 16.92 = 40.92$. So the statement is clearly false.

• Statement 4: The area of the rectangle is 48 + 32√2 cm². The area of a rectangle is calculated as length * width. Let's calculate the area:

  Area = $(8 + 4 \sqrt{2}) * (4 + 2 \sqrt{2})$

  Area = $8 * 4 + 8 * 2 \sqrt{2} + 4 \sqrt{2} * 4 + 4 \sqrt{2} * 2 \sqrt{2}$

  Area = $32 + 16 \sqrt{2} + 16 \sqrt{2} + 16$

  Area = $48 + 32 \sqrt{2}$

  So, the area is indeed $48 + 32 \sqrt{2}$ cm². Therefore, this statement is true! Remember, double-check your calculations, especially with radicals, to avoid any simple errors.

Conclusion: Putting it All Together

So there you have it, guys! We've successfully evaluated all the statements about our rectangle. By understanding the properties of rectangles, applying the Pythagorean Theorem, and carefully calculating, we determined the truth values for each statement. It was a nice little journey through some fundamental geometry and algebra concepts. Always remember to break down problems, use your formulas, and double-check your work!

Here's the summary of our answers:

Keep practicing, keep learning, and keep enjoying the world of math! Until next time, stay curious!