in the isosceles trapezoid ABCD we know: AB||CD,|CD|=c=8cm, height h=7 cm, |<CAB| = 35 degree. Find the area

To find the area of the isosceles trapezoid ABCD, we can use the formula for the area of a trapezoid:

\[
A = \frac{1}{2} \cdot (AB + CD) \cdot h
\]

Where:
- \( AB \) and \( CD \) are the lengths of the parallel sides (bases),
- \( h \) is the height of the trapezoid, which is given as \( h = 7 \, \text{cm} \),
- \( CD = c = 8 \, \text{cm} \),
- \( AB \) is unknown and needs to be determined.

### Step 1: Analyze the geometry
We are given that \( \angle CAB = 35^\circ \). Since ABCD is an isosceles trapezoid and \( AB \parallel CD \), triangles \( \triangle CAB \) and \( \triangle DAB \) are right triangles. The height \( h = 7 \, \text{cm} \) is perpendicular to both \( AB \) and \( CD \).

In \( \triangle CAB \), using trigonometry, we can determine the distance from point \( A \) to the foot of the height, which we’ll call \( x \). This is the horizontal distance that forms part of the base \( AB \).

\[
\tan(35^\circ) = \frac{h}{x}
\]

\[
x = \frac{h}{\tan(35^\circ)} = \frac{7}{\tan(35^\circ)}
\]

### Step 2: Compute \( x \)
Let’s calculate \( x \) to find the horizontal distance from \( A \) to the foot of the height. Then, because the trapezoid is isosceles, the total base \( AB \) will be:

\[
AB = CD + 2x
\]

### Step 3: Calculate the area
Once we have \( AB \), we can substitute it into the formula for the area.

Let me calculate these values.The horizontal distance \( x \) from point \( A \) to the foot of the height is approximately \( 9.997 \, \text{cm} \). 

Thus, the length of the top base \( AB \) is:

\[
AB \approx 27.99 \, \text{cm}
\]

Now, using the formula for the area of the trapezoid, the area is:

\[
A \approx 125.98 \, \text{cm}^2
\]

Would you like more details on the steps, or do you have any further questions?

Here are 5 related questions:
1. How would the area change if the height were doubled?
2. What happens if \( \angle CAB \) were increased to \( 45^\circ \)?
3. Can you find the perimeter of this isosceles trapezoid?
4. How would you calculate the diagonal lengths of this trapezoid?
5. What trigonometric identities are used in solving trapezoid problems?

**Tip:** Understanding the symmetry in isosceles trapezoids simplifies many geometric calculations.

In the isosceles trapezoid ABCD we know: AB || CD, |CD| = c = 8 cm, height h = 7 cm, |<CAB| = 35 degrees. Find the area.

Calculate the area of an isosceles trapezoid with a base of 8 cm, height of 7 cm, and an angle of 35 degrees using trigonometry and geometry. This detailed step-by-step solution is perfect for students in Grades 9-12 learning about trapezoids and trigonometric applications.

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