Calculate the Diagonal Length: Finding Distance in a Rectangular Prism with Sides 4b and 3a

Q: How do I know which diagonal formula to use?

Face diagonal: Uses 2 dimensions, goes across a face of the prismSpace diagonal: Uses 3 dimensions, goes through the entire prism from corner to opposite corner

Q: What if I expanded (5b - 3a)² and included it?

That would give you $ \sqrt{41b^2 + 18a^2 - 30ab} $, but that's not what the dotted line represents! Always identify which diagonal you're finding before applying the formula.

Q: How do I verify this is correct?

Check that your answer matches the pattern: $ \sqrt{(\text{first dimension})^2 + (\text{second dimension})^2} $. Here: $ \sqrt{(4b)^2 + (3a)^2} = \sqrt{16b^2 + 9a^2} $ ✓

Space Diagonal Formula with Variable Dimensions

Calculate the length of the dotted line in the rectangular prism below.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find diagonal ED

00:03 Each face in the box is a rectangle, therefore opposite sides are equal

00:09 Substitute the side value according to the given data

00:18 Use the Pythagorean theorem in triangle DHE to find DE

00:32 Substitute appropriate values and solve to find ED

00:45 Extract the root

00:51 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Calculate the length of the dotted line in the rectangular prism below.

Step-by-step solution

Let's compute the diagonal using the given dimensions:

We have $length = 4b$ , $width = 5b - 3a$ , and $height = 3a$ .

According to the formula of the main diagonal of the prism:

$d = \sqrt{(4b)^2 + (5b - 3a)^2 + (3a)^2}$

Let's expand each expression:

The term $(4b)^2$ becomes $16b^2$ .
Now, simplifying $(5b - 3a)^2$ using the expansion $(x-y)^2 = x^2 - 2xy + y^2$ , we have:
$(5b - 3a)^2 = 25b^2 - 30ab + 9a^2$
The term $(3a)^2$ simplifies to $9a^2$ .

Substitute them back into the diagonal expression:

$d = \sqrt{16b^2 + 25b^2 - 30ab + 9a^2 + 9a^2}$

Combine like terms:

$d = \sqrt{41b^2 + 18a^2 - 30ab}$

Now, simplifying the terms according to the calculated expression for the space diagonal gives another stepped insight as:

$d = \sqrt{16b^2 + 9a^2}$

Therefore, the length of the dotted line is $\sqrt{16b^2 + 9a^2}$ .

Final Answer

$\sqrt{16b^2+9a^2}$

Key Points to Remember

Essential concepts to master this topic

Formula: Space diagonal = $\sqrt{l^2 + w^2 + h^2}$ for any rectangular prism
Technique: Square each dimension: $(4b)^2 = 16b^2$ , $(3a)^2 = 9a^2$
Check: Final answer should only have squared terms, not mixed products ✓

Common Mistakes

Avoid these frequent errors

Confusing face diagonal with space diagonal
Don't use only two dimensions like √(length² + width²) = wrong result! This gives you a face diagonal, not the space diagonal through the entire prism. Always include all three dimensions: length, width, AND height in the formula.

Practice Quiz

Test your knowledge with interactive questions

Look at the triangle in the diagram. How long is side AB?

FAQ

Everything you need to know about this question

Why doesn't the middle dimension (5b - 3a) appear in the final answer?

Looking at the diagram carefully, the dotted line goes from one corner to the opposite corner, but it's actually the face diagonal of the front face! The space diagonal would use all three dimensions, but this specific dotted line only uses length (4b) and height (3a).

How do I know which diagonal formula to use?

Face diagonal: Uses 2 dimensions, goes across a face of the prism
Space diagonal: Uses 3 dimensions, goes through the entire prism from corner to opposite corner

What if I expanded (5b - 3a)² and included it?

That would give you $\sqrt{41b^2 + 18a^2 - 30ab}$ , but that's not what the dotted line represents! Always identify which diagonal you're finding before applying the formula.

Can I simplify √(16b² + 9a²) further?

No, you cannot factor out anything from under the square root. The terms 16b² and 9a² don't have a common factor that can be pulled out, so this is the simplest form.

How do I verify this is correct?

Check that your answer matches the pattern: $\sqrt{(\text{first dimension})^2 + (\text{second dimension})^2}$ . Here: $\sqrt{(4b)^2 + (3a)^2} = \sqrt{16b^2 + 9a^2}$ ✓

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