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We are tasked with finding the area of the shaded triangle \( ABP \) in the given diagram. Here's how we can approach the problem step by step:

### Step 1: Understanding the Triangles
1. **Triangle ABC** is a right-angled triangle.
   - \( CA = 8 \, \text{cm} \) (base)
   - \( PC = 6 \, \text{cm} \), so \( BC = BP + PC = 4 + 6 = 10 \, \text{cm} \).
   - The triangle \( ABC \) is right-angled at \( C \), so we can apply the Pythagorean theorem to find the hypotenuse \( AB \).

### Step 2: Using Pythagoras' Theorem for \( AB \)
For triangle \( ABC \), we have:
\[
AB^2 = AC^2 + BC^2 = 8^2 + 10^2 = 64 + 100 = 164
\]
\[
AB = \sqrt{164} \approx 12.81 \, \text{cm}.
\]

### Step 3: Area of Triangle \( ABC \)
The area of triangle \( ABC \) is:
\[
\text{Area of } ABC = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 8 \times 10 = 40 \, \text{cm}^2.
\]

### Step 4: Area of Triangle \( ABP \)
The area of triangle \( ABP \) is a portion of triangle \( ABC \). Since \( BP = 4 \, \text{cm} \) and \( BC = 10 \, \text{cm} \), the ratio of the height of triangle \( ABP \) to triangle \( ABC \) is:
\[
\frac{BP}{BC} = \frac{4}{10} = 0.4.
\]
Therefore, the area of triangle \( ABP \) is \( 0.4 \) times the area of triangle \( ABC \):
\[
\text{Area of } ABP = 0.4 \times 40 = 16 \, \text{cm}^2.
\]

Thus, the area of the shaded triangle \( ABP \) is **16 cm²**.

Would you like more details or have any further questions?

### Related Questions:
1. How would the area of \( ABP \) change if \( BP \) were 5 cm instead of 4 cm?
2. Can you apply the Pythagorean theorem to non-right-angled triangles?
3. How do you find the area of a triangle if you only know the lengths of its sides?
4. What is the formula for finding the height of a triangle given the base and area?
5. What other methods can be used to find the area of triangle \( ABP \)?

### Tip:
Always check the accuracy of your measurements and ratios when calculating areas of triangles, especially in complex diagrams with multiple triangles.

What is the area of the shaded triangle ABP in the given right-angled triangle ABC, with BP = 4 cm, PC = 6 cm, and CA = 8 cm?

This problem involves calculating the area of the shaded triangle ABP within a right-angled triangle ABC using Pythagoras' theorem and the area formula for triangles. The side lengths provided are BP = 4 cm, PC = 6 cm, and CA = 8 cm. Suitable for Grades 8-10 students.

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