The image you provided shows a triangle with sides $AB = 15$ cm, $AC = 10$ cm, and an angle $\angle CAB = 30^\circ$. The problem asks to find the area of the triangle $\triangle ABC$.

Solution:

To find the area of $\triangle ABC$, we can use the formula for the area of a triangle when two sides and the included angle are known:

$\text{Area} = \frac{1}{2} \times AB \times AC \times \sin(\angle CAB)$

Given:

• $AB = 15$ cm
• $AC = 10$ cm
• $\angle CAB = 30^\circ$

Step 1: Calculate the sine of $30^\circ$.

$\sin(30^\circ) = \frac{1}{2}$

Step 2: Plug in the values into the area formula.

$\text{Area} = \frac{1}{2} \times 15 \times 10 \times \frac{1}{2}$

$\text{Area} = \frac{1}{2} \times 150 \times \frac{1}{2} = \frac{150}{4} = 37.5 \text{ cm}^2$

Final Answer:

The area of the triangle $\triangle ABC$ is $37.5$ square centimeters.

Would you like more details or have any questions? Here are some related questions you might find interesting:

1. How would you find the area of a triangle if all three sides are known?
2. What is the significance of the sine function in calculating the area of a triangle?
3. How would the area change if $\angle CAB$ was $60^\circ$ instead of $30^\circ$?
4. Can the area formula using sine be applied to non-right triangles? Why?
5. How would you calculate the area if only the three angles and one side are known?

Tip: Remember, the sine function for angles greater than $90^\circ$ can be negative, which will affect the calculated area if not handled correctly. Always ensure the angle is within the appropriate range when applying this formula.