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We are given the triangle \( ABC \) with the following details:

- \( BC = 6 \, \text{cm} \)
- \( BD = 22 \, \text{cm} \)
- \( \text{Area of triangle ABC} = 42 \, \text{cm}^2 \)
- \( AC \perp BD \) (i.e., \( AC \) is perpendicular to \( BD \))
- We are tasked to find the size of angle \( x \).

### Step-by-Step Solution:

#### Step 1: Use the area of triangle ABC to find AC.
The area of a right triangle is given by:

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

In triangle \( ABC \), the base is \( BC = 6 \, \text{cm} \) and the height is \( AC \). So we can set up the equation for the area:

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

Simplifying this:

\[
42 = 3 \times AC
\]

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

#### Step 2: Use the Pythagorean theorem to find \( AD \).
Since \( AC \perp BD \), we can break down \( BD \) into two segments:
- \( CD = BD - BC = 22 - 6 = 16 \, \text{cm} \).

Now we have a right triangle \( ACD \) where:
- \( AC = 14 \, \text{cm} \)
- \( CD = 16 \, \text{cm} \)

We can now use the Pythagorean theorem to find \( AD \):

\[
AD^2 = AC^2 + CD^2
\]

Substitute the values:

\[
AD^2 = 14^2 + 16^2 = 196 + 256 = 452
\]

\[
AD = \sqrt{452} \approx 21.26 \, \text{cm}
\]

#### Step 3: Use trigonometry to find angle \( x \).
Now that we have \( AC = 14 \, \text{cm} \) and \( AD = 21.26 \, \text{cm} \), we can find angle \( x \) using the sine rule. In triangle \( ACD \):

\[
\sin(x) = \frac{AC}{AD} = \frac{14}{21.26}
\]

\[
\sin(x) \approx 0.6586
\]

Using the inverse sine function:

\[
x = \sin^{-1}(0.6586) \approx 41.2^\circ
\]

Thus, the size of angle \( x \) is approximately \( 41.2^\circ \).

---

Would you like more details on any of these steps, or have any questions?

Here are 5 related questions for further practice:
1. How do you calculate the area of a triangle given its vertices' coordinates?
2. How can you apply the cosine rule to non-right triangles?
3. What is the difference between sine and cosine rules in solving triangles?
4. How can trigonometric identities help simplify expressions involving angles?
5. How do you calculate angles using tangent in a right-angled triangle?

**Tip:** Always draw a diagram to label known values and visualize the relationships between sides and angles.

Points B, C, and D lie on a straight line. AC is perpendicular to BD. BC = 6 cm and BD = 22 cm. Triangle ABC has an area of 42 cm². Calculate the size of angle x.

Solve a trigonometry and geometry problem involving a right triangle. Given BC = 6 cm, BD = 22 cm, and the area of triangle ABC = 42 cm², calculate the size of angle x using the Pythagorean theorem and sine rule. Suitable for Grades 9-11.

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