Calculate Trapezoid Area: Solving with Height h=(12x-8) and Parallel Sides

Q: Why do I need to add the parallel sides first?

The trapezoid area formula requires the sum of both parallel bases. You can't multiply each base by height separately - you must add $ (x+5) + (13x-2) = 14x+3 $ first!

Q: How do I multiply (14x+3)(12x-8)?

Use the distributive property (FOIL): $ 14x \times 12x = 168x^2 $, $ 14x \times (-8) = -112x $, $ 3 \times 12x = 36x $, $ 3 \times (-8) = -24 $. Then combine like terms.

Q: Why is my answer a quadratic expression?

When you multiply two linear expressions (degree 1), you get a quadratic expression (degree 2). Since both the bases and height contain x, multiplying creates an x² term!

Q: Can I simplify 84x²-38x-12 further?

Check if there's a common factor! Here, 2 divides all terms: $ 84x^2-38x-12 = 2(42x^2-19x-6) $. But the problem asks for the area expression as is.

Q: What if I get confused about which sides are parallel?

In the diagram, the horizontal sides are parallel: the top side $ (x+5) $ and bottom side $ (13x-2) $. The height $ h = 12x-8 $ is always perpendicular to these bases.

Trapezoid Area with Polynomial Expressions

What is the area of the trapezoid in the figure?

$h=(12x-8)$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Express the area of the trapezoid using X

00:03 We'll use the formula for calculating trapezoid area

00:07 (Sum of bases times height) divided by 2

00:12 We'll substitute appropriate values according to the given data, and solve for the area

00:33 Since there's multiplication, we can divide only one factor

00:42 Calculate each quotient

00:49 Open parentheses properly, multiply each factor by each factor

01:11 Calculate the products

01:28 Collect terms

01:31 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

What is the area of the trapezoid in the figure?

$h=(12x-8)$

Step-by-step solution

To solve this problem, we apply the formula for the area of a trapezoid:

Step 1: Identify and substitute the expressions for the bases and height:
$b_1 = x + 5$ , $b_2 = 13x - 2$ , $h = 12x - 8$ .
Step 2: Calculate $b_1 + b_2$ :
$b_1 + b_2 = (x + 5) + (13x - 2) = 14x + 3$ .
Step 3: Substitute into the area formula:
$A = \frac{1}{2} \times (14x + 3) \times (12x - 8)$ .
Step 4: Distribute and simplify:
$A = \frac{1}{2} \times ((14x + 3) \cdot (12x - 8))$ .
Step 5: Expand the product:
$(14x + 3)(12x - 8) = 14x \cdot 12x + 14x \cdot (-8) + 3 \cdot 12x + 3 \cdot (-8) = 168x^2 - 112x + 36x - 24$ .
Step 6: Combine like terms:
$168x^2 - 76x - 24$ .
Step 7: Divide by 2 to find the area:
$A = \frac{1}{2} \times (168x^2 - 76x - 24) = 84x^2 - 38x - 12$ .

Therefore, the area of the trapezoid is $84x^2 - 38x - 12$ .

Final Answer

$84x^2-38x-12$

Key Points to Remember

Essential concepts to master this topic

Formula: Area = ½ × (sum of parallel bases) × height
Technique: First find (x+5)+(13x-2) = 14x+3, then multiply by height
Check: Final answer should be quadratic: 84x²-38x-12 has degree 2 ✓

Common Mistakes

Avoid these frequent errors

Forgetting to divide by 2 in the trapezoid formula
Don't calculate (14x+3)(12x-8) = 168x²-76x-24 as final answer! This gives double the actual area. Always divide the product by 2 since the trapezoid formula is ½ × (b₁+b₂) × h.

Practice Quiz

Test your knowledge with interactive questions

It is possible to use the distributive property to simplify the expression below?

What is its simplified form?

$$ (ab)(c d) $$

$$

FAQ

Everything you need to know about this question

Why do I need to add the parallel sides first?

The trapezoid area formula requires the sum of both parallel bases. You can't multiply each base by height separately - you must add $(x+5) + (13x-2) = 14x+3$ first!

How do I multiply (14x+3)(12x-8)?

Use the distributive property (FOIL): $14x \times 12x = 168x^2$ , $14x \times (-8) = -112x$ , $3 \times 12x = 36x$ , $3 \times (-8) = -24$ . Then combine like terms.

Why is my answer a quadratic expression?

When you multiply two linear expressions (degree 1), you get a quadratic expression (degree 2). Since both the bases and height contain x, multiplying creates an x² term!

Can I simplify 84x²-38x-12 further?

Check if there's a common factor! Here, 2 divides all terms: $84x^2-38x-12 = 2(42x^2-19x-6)$ . But the problem asks for the area expression as is.

What if I get confused about which sides are parallel?

In the diagram, the horizontal sides are parallel: the top side $(x+5)$ and bottom side $(13x-2)$ . The height $h = 12x-8$ is always perpendicular to these bases.

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