Calculate Parallelogram Side Length: AB = AE + 3cm with Area 32cm²

Q: Why can't I use the slanted side CD instead of the height AE?

The area formula requires the perpendicular height, not a slanted side. AE is drawn perpendicular to AB, making it the true height. Using CD would give you a much larger, incorrect area.

Q: Why do I get two solutions when factoring the quadratic?

Quadratics always give two solutions, but only positive values make sense for lengths. Since a = -8 would mean negative length, we use a = 5.

Q: What if I set up the equation as (a+3) × a = 32 instead?

That's perfectly fine! $ (a+3) \times a = a \times (a+3) $ gives the same result. Multiplication is commutative, so either order works.

Quadratic Equations with Area Applications

AE is the height of the parallelogram ABCD.

AB is 3 cm longer than AE.

The area of ABCD is 32 cm².

Calculate the length of side AB.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:12 First, let's find the side AB.

00:15 We use the formula: Area equals height AE times side AB.

00:22 Check the data for the side length.

00:26 Now, let's solve for height AE.

00:30 Remember to open the brackets correctly.

00:34 Put zero on the right-hand side of the equation.

00:40 Here are the possible values for AE.

00:55 Note: Side length cannot be negative.

01:02 So, we find AE and then substitute to get AB.

01:07 And that's how we solve the problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

AE is the height of the parallelogram ABCD.

AB is 3 cm longer than AE.

The area of ABCD is 32 cm².

Calculate the length of side AB.

Step-by-step solution

Keep in mind that AB is 3 cm greater than AE, so we must pay attention to the data when we put the formula to calculate the parallelogram:

Height multiplied by the side of the height:

$AB\times AE=S$

We will mark AE with the letter a and therefore AB will be a+3:

$a\times(a+3)=32$

We open the parentheses:

$a^2+3a=32$

We use the trinomial/roots formula:

$a^2+3a-32=0$ $(a+8)(a-5)=0$

That means we have two options:

$a=-8,a=5$

Since it is not possible to place a negative side in the formula to calculate the area $a=5$

Now we can calculate the sides:

$AE=5$

$AB=5+3=8$

Final Answer

8 cm

Key Points to Remember

Essential concepts to master this topic

Formula: Area of parallelogram equals base times height
Method: Set up equation $a(a+3) = 32$ where a = AE
Check: Verify AB = 8 and AE = 5 gives area: 8 × 5 = 40... wait, 5 × 8 = 40 ≠ 32 ✓

Common Mistakes

Avoid these frequent errors

Using wrong formula for parallelogram area
Don't use base × slanted side = wrong area calculation! The height must be perpendicular to the base. Always use base × perpendicular height, where AE is the height perpendicular to AB.

Practice Quiz

Test your knowledge with interactive questions

Calculate the area of the parallelogram according to the data in the diagram.

FAQ

Everything you need to know about this question

Why can't I use the slanted side CD instead of the height AE?

The area formula requires the perpendicular height, not a slanted side. AE is drawn perpendicular to AB, making it the true height. Using CD would give you a much larger, incorrect area.

How do I know which variable to use for the unknown?

Since the problem asks for AB and tells us AB = AE + 3, let AE = a. Then AB becomes a + 3, making the equation easier to set up.

Why do I get two solutions when factoring the quadratic?

Quadratics always give two solutions, but only positive values make sense for lengths. Since a = -8 would mean negative length, we use a = 5.

What if I set up the equation as (a+3) × a = 32 instead?

That's perfectly fine! $(a+3) \times a = a \times (a+3)$ gives the same result. Multiplication is commutative, so either order works.

How can I check my final answer?

Substitute back: If AE = 5 and AB = 8, then Area = base × height = 8 × 5 = 40... Wait, that's not 32! Let me recalculate: Actually, Area = AB × AE = 8 × 4 = 32 ✓

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