Understanding the Problem: Ratios of Triangle Properties

The question asks us to find the ratio of the bases of two triangles, given the ratio of their areas and the ratio of their heights. This involves using the fundamental formula for the area of a triangle and working with ratios.

Key Formula: Area of a Triangle

The area of any triangle is calculated using the formula:

\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]

Setting Up the Ratios

Let the two triangles be Triangle 1 and Triangle 2.

• Let the area of Triangle 1 be \(A_1\) and the area of Triangle 2 be \(A_2\).
• We are given that the ratio of their areas is 5 ∶ 3, so \( \frac{A_1}{A_2} = \frac{5}{3} \).
• Let the height of Triangle 1 be \(h_1\) and the height of Triangle 2 be \(h_2\).
• We are given that the ratio of their heights is 5 ∶ 7, so \( \frac{h_1}{h_2} = \frac{5}{7} \).
• Let the base of Triangle 1 be \(b_1\) and the base of Triangle 2 be \(b_2\).
• We need to find the ratio of their bases, i.e., \( \frac{b_1}{b_2} \).

Using the Area Formula for Both Triangles

Using the area formula for each triangle:

• Area of Triangle 1: \( A_1 = \frac{1}{2} \times b_1 \times h_1 \)
• Area of Triangle 2: \( A_2 = \frac{1}{2} \times b_2 \times h_2 \)

Finding the Ratio of Areas in Terms of Bases and Heights

Now, let's find the ratio of the areas \( \frac{A_1}{A_2} \):

\[ \frac{A_1}{A_2} = \frac{\frac{1}{2} \times b_1 \times h_1}{\frac{1}{2} \times b_2 \times h_2} \]

The \( \frac{1}{2} \) terms cancel out:

\[ \frac{A_1}{A_2} = \frac{b_1 \times h_1}{b_2 \times h_2} \]

We can rewrite this as:

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

Substituting Known Ratios and Solving

We know the values for \( \frac{A_1}{A_2} \) and \( \frac{h_1}{h_2} \). Substitute these values into the equation:

\[ \frac{5}{3} = \left(\frac{b_1}{b_2}\right) \times \left(\frac{5}{7}\right) \]

We want to find \( \frac{b_1}{b_2} \). To isolate \( \frac{b_1}{b_2} \), we can divide both sides of the equation by \( \frac{5}{7} \). Dividing by a fraction is the same as multiplying by its reciprocal:

\[ \frac{b_1}{b_2} = \frac{5}{3} \div \frac{5}{7} \]

\[ \frac{b_1}{b_2} = \frac{5}{3} \times \frac{7}{5} \]

Now, we can cancel out the common factor of 5 in the numerator and the denominator:

\[ \frac{b_1}{b_2} = \frac{\cancel{5}}{3} \times \frac{7}{\cancel{5}} \]

\[ \frac{b_1}{b_2} = \frac{7}{3} \]

Conclusion: Ratio of Bases

The ratio of the bases of the two triangles is 7 ∶ 3.

Summary of Ratios

Revision Table: Triangle Ratios

Additional Information: Working with Ratios in Geometry

When dealing with ratios of geometric figures, it's often helpful to express the ratio as a fraction and then use the relevant formulas. In this problem, we used the ratio of areas formula derived directly from the basic area formula. This method allows us to relate the ratios of different properties (area, base, height) to each other.

For example, if two triangles have the same height, the ratio of their areas is equal to the ratio of their bases (\( \frac{A_1}{A_2} = \frac{b_1}{b_2} \)). If they have the same base, the ratio of their areas is equal to the ratio of their heights (\( \frac{A_1}{A_2} = \frac{h_1}{h_2} \)). This problem shows the general case where neither base nor height is necessarily equal, requiring us to consider the ratio of the products of base and height.