In figure below, $∠\ ABC\ =\ 90^o$ and $BD\ ⊥\ AC$. If $BD\ =\ 8\ cm$, and $AD\ =\ 4\ cm$, find $CD$.
"
Given:
In the given figure $∠\ ABC\ =\ 90^o$ and $BD\ ⊥\ AC$.
$BD\ =\ 8\ cm$, and $AD\ =\ 4\ cm$.
To do:
We have to find $CD$.
Solution:
In $\vartriangle ABD$ and $\vartriangle BCD$,
$\angle ABD+\angle DBC=90^o$
$\angle C+\angle DBC=90^o$
This implies,
$\angle ABD=\angle C$
In $\vartriangle ABD$ and $\vartriangle BCD$,
$\angle ABD=\angle C$
$\angle ADB=\angle BDC=90^o$
Therefore,
$\vartriangle DBA∼\vartriangle DCB$ (By AA similarity)
$\frac{BD}{CD} = \frac{AD}{BD}$ (Corresponding parts of similar triangles are proportional)
$BD^2 = AD \times DC$
$(8)^2 = 4 \times DC$
$DC = \frac{64}{4} = 16\ cm$
The measure of $CD$ is $16\ cm$.
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