Question : The areas of the two triangles are in the ratio 4 : 3 and their heights are in the ratio 6 : 5. Find the ratio of their bases.
Option 1: 5 : 6
Option 2: 10 : 9
Option 3: 6 : 5
Option 4: 9 : 10
Correct Answer: 10 : 9
Solution :
The area of a triangle is given by the formula $\frac{1}{2} \times \text{base} \times \text{height}$.
Given that the areas of the two triangles are in the ratio 4 : 3 and their heights are in the ratio 6 : 5.
$\frac{\frac{1}{2} \times \text{base}_1 \times \text{height}_1}{\frac{1}{2} \times \text{base}_2 \times \text{height}_2} = \frac{4}{3}$
Substituting the given ratio of the heights into this equation gives:
$\frac{\text{base}_1 \times 6}{\text{base}_2 \times 5} = \frac{4}{3}$
Solving this equation for the ratio of the bases gives:
$\frac{\text{base}_1}{\text{base}_2} = \frac{4}{3} \times \frac{5}{6} = \frac{20}{18} = \frac{10}{9}$
Hence, the correct answer is 10 : 9.
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