Cho tam giác ABC, M là điểm nằm trên cạnh BC. N là điểm nằm trên cạnh AC sao cho: Diện tích tam giác ABM = 30 cm²? Diện tích tam giác ABN

To solve the problem for triangle \( ABC \) given the conditions related to points \( M \) and \( N \), we will analyze the information step-by-step.

### Given:
- \( \text{Area of } \triangle ABM = 30 \, \text{cm}^2 \)
- \( AN = \frac{2}{5} AC \)

### Part a: Find the area of triangle \( ABC \)

1. **Using ratios of areas**:
- Area ratios of triangles sharing a common vertex and base segment:
\[
\frac{[ABM]}{[ABC]} = \frac{BM}{BC}
\]
Since \( M \) is the midpoint of \( BC \), \( BM = MC \). Thus, \( BC = 2BM \).

Now, we have:
\[
\frac{[ABM]}{[ABC]} = \frac{BM}{2BM} = \frac{1}{2}
\]

2. **Calculating area of triangle ABC**:
\[
[ABC] = 2 \times [ABM] = 2 \times 30 = 60 \, \text{cm}^2
\]

### Part b: Find the area of triangle \( ABN \)

1. **Using the ratio \( AN = \frac{2}{5} AC \)**:
- The area of triangle \( ABN \) can be expressed as:
\[
\frac{[ABN]}{[ABC]} = \frac{AN}{AC}
\]
Since \( AN = \frac{2}{5} AC \), we have:
\[
\frac{[ABN]}{[ABC]} = \frac{2/5 \, AC}{AC} = \frac{2}{5}
\]

2. **Calculating the area of triangle ABN**:
\[
[ABN] = \frac{2}{5} \times [ABC] = \frac{2}{5} \times 60 = 24 \, \text{cm}^2
\]

### Summary of the Results:
- **Area of triangle \( ABC \)**: \( 60 \, \text{cm}^2 \)
- **Area of triangle \( ABN \)**: \( 24 \, \text{cm}^2 \)