Calculate Edge Length AE in a Rectangular Prism Using √(26a²+8a+16)

Q: Why does EF equal HG in this problem?

In a rectangular prism, opposite edges are equal. Since EFGH forms one rectangular face and ABCD forms the opposite face, corresponding edges like EF and HG have the same length.

Q: How do I expand (a + 4)² correctly?

Use the formula $ (x + y)^2 = x^2 + 2xy + y^2 $. So $ (a + 4)^2 = a^2 + 8a + 16 $.

Q: What if I get a negative value under the square root?

Check your algebra! In this problem, when you subtract $ a^2 + 8a + 16 $ from $ 26a^2 + 8a + 16 $, you should get $ 25a^2 $, which is always positive.

Q: Why is the answer 5a and not 25a?

Remember: $ AE = \sqrt{(AE)^2} = \sqrt{25a^2} $. Since $ \sqrt{25a^2} = \sqrt{25} \cdot \sqrt{a^2} = 5|a| $, and we assume a > 0 for length, AE = 5a.

Pythagorean Theorem with Algebraic Expressions

$ABCDEFGH$ is a rectangular prism.

$AF=\sqrt{26a^2+8a+16}$

$HG=A+4$

Calculate $AE$ .

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Calculate AE

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

00:11 We'll use the Pythagorean theorem in triangle AEF to find AE

00:25 We'll substitute appropriate values according to the given data and solve for AE

00:39 We'll expand brackets properly

00:47 We'll isolate AE, reduce what we can and arrange the equation

01:09 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

$ABCDEFGH$ is a rectangular prism.

$AF=\sqrt{26a^2+8a+16}$

$HG=A+4$

Calculate $AE$ .

Step-by-step solution

To solve this problem of calculating $AE$ in the rectangular prism, we will use the Pythagorean Theorem:

Step 1: Identify given variables:
We know $AF = \sqrt{26a^2 + 8a + 16}$ and $HG = a + 4$ .

Step 2: Problem Setup:
We recognize that $AF$ is a diagonal across the face $AFE$ in the prism. Given expressions can be linked: $(AE)^2 + (EF)^2 = (AF)^2$ .

Step 3: Utilize $HG$ :
Since $HG = a + 4$ , $EF = HG = a + 4$ as triangles and prisms share dimensions proportionally, so $(AE)^2 + (a + 4)^2 = 26a^2 + 8a + 16$ .

Step 4: Equation Simplification:
We'll expand and simplify further:
$(AE)^2 + a^2 + 8a + 16 = 26a^2 + 8a + 16.$
Solving gives us $(AE)^2 = 25a^2$ .

Step 5: Solution Conclusion:
Calculate $AE$ as $\sqrt{25a^2} = 5a$ , through recognizing $AE$ only needs formula $\sqrt{x}$ operation once simplified.

Therefore, the calculated length $AE$ is $5a$ .

Final Answer

$5a$

Key Points to Remember

Essential concepts to master this topic

Space Diagonal Rule: In rectangular prisms, use 3D Pythagorean theorem
Technique: Set up $(AE)^2 + (EF)^2 = (AF)^2$ with given values
Check: Verify $(5a)^2 + (a+4)^2 = 26a^2 + 8a + 16$ ✓

Common Mistakes

Avoid these frequent errors

Confusing which edges form the right triangle
Don't randomly pick three edges to form a triangle = wrong setup! The diagonal AF connects opposite vertices, so you need the two perpendicular edges that meet at A. Always identify which face or space the diagonal crosses.

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

How do I know which edges to use in the Pythagorean theorem?

Look at the diagonal AF - it goes from vertex A to vertex F. The two perpendicular edges meeting at A are AE (what we want) and another edge. Since EFGH forms a rectangle, EF = HG = a + 4.

Why does EF equal HG in this problem?

In a rectangular prism, opposite edges are equal. Since EFGH forms one rectangular face and ABCD forms the opposite face, corresponding edges like EF and HG have the same length.

How do I expand (a + 4)² correctly?

Use the formula $(x + y)^2 = x^2 + 2xy + y^2$ . So $(a + 4)^2 = a^2 + 8a + 16$ .

What if I get a negative value under the square root?

Check your algebra! In this problem, when you subtract $a^2 + 8a + 16$ from $26a^2 + 8a + 16$ , you should get $25a^2$ , which is always positive.

Why is the answer 5a and not 25a?

Remember: $AE = \sqrt{(AE)^2} = \sqrt{25a^2}$ . Since $\sqrt{25a^2} = \sqrt{25} \cdot \sqrt{a^2} = 5|a|$ , and we assume a > 0 for length, AE = 5a.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Pythagorean Theorem questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime