Discussion

Explanation:

In right triangle ABC, since AB and BC have to be integers it has to be a pythagorean triplet.
Since, AC = 17, the applicable pythagorean triplet is 8, 15 and 17.
Now, (AB, BC, AC) = (8, 15 and 17) or (15, 8 and 17).

In ∆ABC and ∆BCD
∠ABC = ∠BCD, &
∠BAC = ∠CBD
∴ ∆ABC is similar to ∆BCD

Case 1: (AB, BC, AC) = (8, 15 and 17)
$\frac{\mathrm{Area} \mathrm{of} ∆\mathrm{ABC}}{\mathrm{Area} \mathrm{of} ∆\mathrm{BCD}}$ = ${\left(\frac{\mathrm{AB}}{\mathrm{BC}}\right)}^2$ = ${\left(\frac{8}{15}\right)}^2$ = $\frac{64}{225}$

Case 2: (AB, BC, AC) = (15, 8 and 17)
$\frac{\mathrm{Area} \mathrm{of} ∆\mathrm{ABC}}{\mathrm{Area} \mathrm{of} ∆\mathrm{BCD}}$ = ${\left(\frac{\mathrm{AB}}{\mathrm{BC}}\right)}^2$ = ${\left(\frac{15}{8}\right)}^2$ = $\frac{225}{64}$

Hence, option (d).

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