https://file.aiquickdraw.com/mathbot/file/1721783852635jexu4ay7.jpg

To find the area of the trapezoid, we'll use the formula for the area of a trapezoid:

\[ A = \frac{1}{2} \times (b_1 + b_2) \times h \]

where \( b_1 \) and \( b_2 \) are the lengths of the two parallel sides (bases) of the trapezoid, and \( h \) is the height.

In the provided image, the bases of the trapezoid are 6 and 12 units. To find the height, we can use the properties of the trapezoid and the Pythagorean theorem. The trapezoid is isosceles, with each non-parallel side being 6 units.

Here are the steps:

1. Draw the height from the top vertices perpendicular to the base. This will form two right triangles and a rectangle in the middle.
2. The length of the rectangle's base is the difference between the length of the larger base and the smaller base, divided by 2: 
\[ \frac{12 - 6}{2} = 3 \] units.

3. Now, we have a right triangle with the hypotenuse of 6 units and one leg of 3 units. To find the height (other leg) \( h \):

\[ h = \sqrt{6^2 - 3^2} = \sqrt{36 - 9} = \sqrt{27} = 3\sqrt{3} \]

Now we have all the necessary components to find the area:

\[ A = \frac{1}{2} \times (6 + 12) \times 3\sqrt{3} \]
\[ A = \frac{1}{2} \times 18 \times 3\sqrt{3} \]
\[ A = 9 \times 3\sqrt{3} \]
\[ A = 27\sqrt{3} \]

So, the area of the trapezoid is \( 27\sqrt{3} \) square units.

Would you like more details or have any questions?

Here are 5 more questions you might want to explore:
1. How do you find the perimeter of a trapezoid?
2. What is the area of a trapezoid with bases of 8 units and 14 units and a height of 5 units?
3. How can the area formula for a trapezoid be derived?
4. What are the properties of an isosceles trapezoid?
5. How can you determine the height of a trapezoid if the area and bases are known?

**Tip:** Always make sure to check if the trapezoid is isosceles, as it simplifies calculations by allowing you to use the properties of right triangles.

Learn how to find the area of a trapezoid with bases of 6 and 12 units and a height of 3√3 units using the area formula for trapezoids. This problem involves geometric concepts like trapezoids, isosceles trapezoids, and applications of the Pythagorean theorem. Suitable for high school students.

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