https://file.aiquickdraw.com/mathbot/file/1726091628984c1lihwsk.jpg

The problem in the image involves calculating the angle \( x \) in a triangle.

### Given information:
- Points \( B \), \( C \), and \( D \) lie on a straight line.
- \( AC \perp BD \) (i.e., \( AC \) is perpendicular to \( BD \)).
- \( BC = 6 \, \text{cm} \), \( AD = 30 \, \text{cm} \).
- The area of triangle \( ABC \) is \( 42 \, \text{cm}^2 \).

### Step 1: Use the area of triangle formula.
The area of a triangle is given by the formula:

\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]

In triangle \( ABC \), \( BC \) is the base and \( AC \) is the height. So, we have:

\[
42 = \frac{1}{2} \times 6 \times AC
\]

Solving for \( AC \):

\[
42 = 3 \times AC
\]
\[
AC = \frac{42}{3} = 14 \, \text{cm}
\]

So, the height \( AC \) is 14 cm.

### Step 2: Use trigonometry to find angle \( x \).
Now, we need to find angle \( x \) in triangle \( ACD \). In this right triangle, we have:
- \( AD = 30 \, \text{cm} \) (adjacent side),
- \( AC = 14 \, \text{cm} \) (opposite side).

We can use the tangent function:

\[
\tan(x) = \frac{\text{opposite}}{\text{adjacent}} = \frac{AC}{AD} = \frac{14}{30}
\]

\[
\tan(x) = \frac{7}{15}
\]

Now, find the value of \( x \) using the inverse tangent function:

\[
x = \tan^{-1}\left(\frac{7}{15}\right)
\]

Let me calculate that for you.The size of angle \( x \) is approximately \( 25.02^\circ \).

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

Here are 5 related questions to consider:
1. How would you calculate angle \( x \) if the area of triangle \( ABC \) were different?
2. Can you solve for angle \( x \) using a different trigonometric function?
3. How would this problem change if triangle \( ABC \) was not a right triangle?
4. How do we use the sine or cosine rule to solve non-right triangles?
5. How would you calculate the length of side \( CD \) in this triangle?

**Tip:** When dealing with right triangles, always consider which trigonometric ratio is most appropriate based on the sides given (opposite, adjacent, or hypotenuse).

Points B, C, and D lie on a straight line. AC is perpendicular to BD. BC = 6 cm and AD = 30 cm. Triangle ABC has an area of 42 cm². Calculate the size of angle x.

Learn how to calculate the angle x in a right triangle with perpendicular sides, given the area, side lengths, and using trigonometric functions. This step-by-step solution includes formulas for area and tangent functions. Ideal for Grade 9-10 students.

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