Value of x for the similar triangles ΔABD ~ ΔCAD is equals to [tex]\frac{25}{4}[/tex].
What are similar triangles?
" Similar triangles are defined as the triangles with same shape but different in their size, corresponding sides of similar triangles are always in proportion."
According to the question,
As per the diagram,
ΔABD and ΔCAD are similar triangles.
AB = 10 units
BD = x units
BC = 16 units
CD = BC - BD
= 16 - x
ΔABD is a right angle triangle.
Therefore,
Using Pythagoras theorem,
[tex]AD ^{2} = 10^{2} -x^{2}[/tex]
ΔABD corresponding ΔCAD as per the given diagram.
From the definition of similar triangles corresponding sides are in proportion.
[tex]\frac{AD}{CD} =\frac{BD}{AD}[/tex]
⇒[tex]AD^{2} =(BD )(CD)[/tex]
Substitute the value of AD, BD and CD we get,
[tex]10^{2} -x^{2} = (x) (16-x)[/tex]
⇒[tex]100 -x^{2} =16x-x^{2}[/tex]
⇒[tex]100 = 16x[/tex]
⇒[tex]x= \frac{100}{16}[/tex]
⇒[tex]x=\frac{25}{4}[/tex]
Hence, Option(F) is the correct answer.
Learn more about similar triangles here
https://brainly.com/question/25882965
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