Triangle Base Ratio Calculation

The area ($A$) of a triangle is given by the formula:

$A = 1/2 × b × h$

where $b$ represents the base and $h$ represents the height.

Deriving the Ratio of Bases

Let the areas of the two triangles be $A_1$ and $A_2$, their bases be $b_1$ and $b_2$, and their heights be $h_1$ and $h_2$.

We are given the following ratios:

• Area ratio: $ \frac{A_1}{A_2} = \frac{4}{3} $
• Height ratio: $ \frac{h_1}{h_2} = \frac{3}{4} $

The ratio of the areas can be written as:

$ \frac{A_1}{A_2} = \frac{ (1/2) \times b_1 \times h_1 }{ (1/2) \times b_2 \times h_2 } $

Simplifying this, we get:

$ \frac{A_1}{A_2} = \left( \frac{b_1}{b_2} \right) \times \left( \frac{h_1}{h_2} \right) $

Now, substitute the given values into the equation:

$ \frac{4}{3} = \left( \frac{b_1}{b_2} \right) \times \left( \frac{3}{4} \right) $

To find the ratio of the bases ($ \frac{b_1}{b_2} $), we rearrange the equation:

$ \frac{b_1}{b_2} = \frac{4}{3} \div \frac{3}{4} $

Performing the division:

$ \frac{b_1}{b_2} = \frac{4}{3} \times \frac{4}{3} $

$ \frac{b_1}{b_2} = \frac{16}{9} $

Thus, the ratio of the bases is 16 : 9.