Question : In $\triangle $ABC, AD$\perp$ BC and AD2 = BD × DC. The measure of $\angle$ BAC is:
Option 1: 60°
Option 2: 75°
Option 3: 90°
Option 4: 45°
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Correct Answer: 90°
Solution :
Given: In $\triangle$ABC, AD$\perp$BC and AD
2
= BD × DC.
Applying Pythagoras theorem in $\triangle$ADB and $\triangle$ADC,
AB
2
= AD
2
+ BD
2
____(i)
AC
2
= AD
2
+ DC
2
____(ii)
Adding equation (i) and (ii),
AB
2
+ AC
2
= AD
2
+ BD
2
+ AD
2
+ DC
2
AB
2
+ AC
2
= 2AD
2
+ BD
2
+ DC
2
AB
2
+ AC
2
= 2 BD × DC + BD
2
+ DC
2
AB
2
+ AC
2
= (BD + DC)
2
AB
2
+ AC
2
= BC
2
$\triangle$ABC is a right-angled triangle.
$\angle$BAC = 90°
Hence, the correct answer is 90°.
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