In the given figure, $∠ A C B$ =90° and $C D ⊥ A B$ . State if the given condition true or false “ $C B 2 C A 2 = B D A D$ ”.

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False

True

answer is B.

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Detailed Solution

It is given that, ∠ACB=

90^{°}

and CD⊥AB.
Question Image

If a perpendicular is drawn from the right angle's vertex to the hypotenuse, the triangle on both sides of the perpendicular is similar to the whole triangle and to each other.
We can observe from the figure that in triangle ACD and ABC,
∠ADC=∠ACB(90°)
∠CAD=∠CAB(same)
Triangle ACD

\approx Δ

ABC by AA criterion.
In triangle BCD and BAC
∠BDC=∠BCA(90°)
∠CBD=∠CBA(same)
Hence, the triangle BCD

\approx

BAC by AA criterion.
We can observe from the figure that, ΔACD~ΔABC and ΔBCD~ΔBAC with AA similarity criterion.
Since, corresponding sides of similar triangles are proportional, we can write that,