If two triangles are on the same base and between the same parallels. Then find the ratio of area of the two triangles.

Answer

To find the ratio of the areas of two triangles that are on the same base and between the same parallels, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Triangles**: Let’s denote the two triangles as Triangle 1 (Δ1) and Triangle 2 (Δ2). Both triangles share the same base and are situated between two parallel lines.

2. **Understand the Base**: Since both triangles are on the same base, we can denote the length of the base as \( b \). Therefore, for both triangles:
\[ \text{Base of } Δ1 = \text{Base of } Δ2 = b \]

3. **Determine the Heights**: The height of a triangle is the perpendicular distance from the base to the opposite vertex. Since both triangles are between the same parallels, the heights (h1 for Triangle 1 and h2 for Triangle 2) will be equal. We can denote:
\[ h_1 = h_2 = h \]

4. **Use the Area Formula for Triangles**: The area \( A \) of a triangle is given by the formula:
\[ A = \frac{1}{2} \times \text{Base} \times \text{Height} \]
Therefore, the area of Triangle 1 (A1) and Triangle 2 (A2) can be expressed as:
\[ A_1 = \frac{1}{2} \times b \times h_1 \]
\[ A_2 = \frac{1}{2} \times b \times h_2 \]

5. **Set Up the Ratio of Areas**: The ratio of the areas of the two triangles is:
\[ \frac{A_1}{A_2} = \frac{\frac{1}{2} \times b \times h_1}{\frac{1}{2} \times b \times h_2} \]

6. **Simplify the Ratio**: Cancel out the common terms (1/2 and b):
\[ \frac{A_1}{A_2} = \frac{h_1}{h_2} \]

7. **Substitute Heights**: Since \( h_1 = h_2 \):
\[ \frac{A_1}{A_2} = \frac{h}{h} = 1 \]

8. **Conclusion**: Therefore, the ratio of the areas of the two triangles is:
\[ \text{Ratio of areas} = 1:1 \]