In the diagram above, triangle ABC has a right angle at B and a perime

Let us see what we are asked.
Area of ABD=\(\frac{1}{2}*AD*BD\)
Area of CBD=\(\frac{1}{2}*CD*BD\)
Thus their ratio = Area of ABD : Area of CBD =\(\frac{1}{2}*AD*BD\): \(\frac{1}{2}*CD*BD\)=AD:CD
So what we have in options is AD/CD

Next, triangles ABD and CBD, also triangle ABC, are similar, and we will get AD*CD=BD^2.
That is so because, triangles ABD and CBD give us \(\frac{AD}{BD}=\frac{BD}{CD}\)

Thus, \(AD*CD=BD^2=24^2\)

Various ways to solve further

1) AD and CD have to be factors of 24^2, so AD/CD too should have only factors of 24^2. ONLY 16/9 possible.

2) The ratio AD/CD should come out as some square/some square. Again 16/9, which is 4^2/3^2 fits in.

3) We can take the help of perimeter given to solve.


D

in #3 how can we use the perimeter to get the ratio?

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