Question

In ΔABC, m∠B=90 and BD is an altitude. The correspondence BDA ⇔...... between ΔBDA and ΔBDC is a similarity.Fill in the blank so that the given statement is true.

Answer 1

answer :- CDB

explanation :- see figure , drawn a triangle ABC in such a way that m∠B=90° and BD is altitude on AC.

Given: In ∆ABC,
∠ABC = ∠ADB = ∠BDC = 90°
In ∆BDA and ∆CDB,
∠BDA = ∠CDB = 90°
∠ABD = ∠BCD [measure of both being equal, i.e., (90° - m∠A)]
This can be explained as,
In ∆BDA, by angle sum property of triangle,
∠BDA + ∠DAB + ∠ABD = 180°
⇒ 90° + ∠A + ∠ABD = 180° [∵, ∠BDA = 90° &∠DAB = ∠A]
⇒ ∠ABD = 180° - 90° - ∠A
⇒ ∠ABD = 90° - ∠A …(i)
Similarly, in ∆ABC by angle sum property of triangle,
∠ABC + ∠BAC + ∠BCA = 180°
⇒ 90° + ∠A + ∠BCD = 180° [∵, ∠ABC = 90°,∠BAC = ∠A & ∠BCA = ∠BCD (from the figure)]
⇒ ∠BCD = 180° - 90° - ∠A
⇒ ∠BCD = 90° - ∠A …(ii)
By equations (i) and (ii), we get
∠ABD = ∠BCD
Hence, by AA corollary the correspondence BDA ⇔ CDB is similarity between ∆BDA and∆BDC.
Thus, the answer is CDB.