in the isosceles trapezoid abcd we know ab is parallel to cd and c is equal to 8 cm, height is 7 cm and angle cab is 35 degrees. find the area of the trapezoid

To find the area of an isosceles trapezoid \( ABCD \), we can use the following formula for the area \( A \) 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 7 cm.

### Step 1: Identify given values
- \( CD = 8 \, \text{cm} \),
- \( h = 7 \, \text{cm} \),
- \( \angle CAB = 35^\circ \).

### Step 2: Determine the length of \( AB \)

We know that in an isosceles trapezoid, the non-parallel sides \( AD \) and \( BC \) are congruent. The angle \( \angle CAB \) allows us to use trigonometry to find the additional information needed to calculate the length of \( AB \).

From point \( A \), drop a perpendicular to the base \( CD \) at a point we'll call \( E \). This creates a right triangle \( ABE \) where:
- \( BE \) is half the difference between \( AB \) and \( CD \),
- \( AE = h = 7 \, \text{cm} \).

Now, using trigonometry, we can calculate \( BE \) as follows:
\[
\tan(\angle CAB) = \frac{BE}{AE}
\]

Substituting the values:
\[
\tan(35^\circ) = \frac{BE}{7}
\]

Solving for \( BE \):
\[
BE = 7 \cdot \tan(35^\circ)
\]

### Step 3: Calculate the value of \( BE \)

Let's compute \( BE \):

\[
BE \approx 7 \cdot 0.7002 \approx 4.901 \, \text{cm}
\]

Thus, \( BE \approx 4.9 \, \text{cm} \).

### Step 4: Determine the length of \( AB \)

Since \( BE \) is half the difference between \( AB \) and \( CD \), we have:
\[
AB = CD + 2 \cdot BE = 8 + 2 \cdot 4.9 = 8 + 9.8 = 17.8 \, \text{cm}
\]

### Step 5: Calculate the area of the trapezoid

Now that we know the lengths of the parallel sides \( AB = 17.8 \, \text{cm} \) and \( CD = 8 \, \text{cm} \), we can use the area formula:

\[
A = \frac{1}{2} \cdot (17.8 + 8) \cdot 7 = \frac{1}{2} \cdot 25.8 \cdot 7 = 90.3 \, \text{cm}^2
\]

Thus, the area of the trapezoid is approximately \( 90.3 \, \text{cm}^2 \).

Would you like more details or have any questions?

### Related Questions:
1. How do we derive the area formula for a trapezoid?
2. How would the problem change if the angle \( CAB \) was 45 degrees instead of 35 degrees?
3. What happens if the trapezoid is not isosceles? How would you calculate the area then?
4. Can you find the lengths of the non-parallel sides \( AD \) and \( BC \)?
5. How would the solution change if the height was given as 10 cm?

### Tip:
Using trigonometric functions is a powerful method to solve geometry problems involving angles and side lengths.

In the isosceles trapezoid ABCD, we know AB is parallel to CD and CD is equal to 8 cm, height is 7 cm, and angle CAB is 35 degrees. Find the area of the trapezoid.

Learn how to calculate the area of an isosceles trapezoid ABCD where AB is parallel to CD, CD = 8 cm, the height is 7 cm, and angle CAB is 35 degrees. The problem involves trigonometry and geometry, using key formulas such as the trapezoid area formula and the tangent function. Suitable for students in Grades 9-11.

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