Question : D is a point on the side BC of a $\triangle $ABC such that $\angle A D C=\angle B A C$. If CA = 10 cm and BC=16 cm, then the length of CD is:
Option 1: 6.5 cm
Option 2: 6.25 cm
Option 3: 7 cm
Option 4: 6 cm
Correct Answer: 6.25 cm
Solution :
In this case, we can use the property of similar triangles to find the length of CD.
Given that $\angle ADC = \angle BAC$, $\triangle ADC$ is similar to $\triangle BAC$ by the AAA similarity criterion.
This implies that the ratios of the corresponding sides of these triangles are equal. Therefore,
$⇒\frac{CA}{CD} = \frac{CB}{CA}$
Rearranging the terms,
$⇒CA^2 = CB \times CD$
Substituting the given values $CA = 10$ cm and $CB = 16$ cm
$⇒100 = 16 \times CD$
$⇒CD = \frac{100}{16} = 6.25 \text{ cm}$
Hence, the correct answer is 6.25 cm.
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