In triangle ABC, AD _I_ BC. Such that AD2 = BD x CD. Prove that triangle ÁBC is right
angled triangle.
given figure
Answers
Given : In triangle ABC , AD is perpendicular to BC and AD² = BD.DC
To prove : BAC = 90°
Proof : in right triangles ∆ADB and ∆ADC
So, Pythagoras theorem should be apply ,
Then we have ,
AB² = AD² + BD² ----------(1)
AC²= AD²+ DC² ---------(2)
AB² + AC² = 2AD² + BD²+ DC²
= 2BD . CD + BD² + CD² [ ∵ given AD² = BD.CD ]
= (BD + CD )² = BC²
Thus in triangle ABC we have , AB² + AC²= BC²
hence triangle ABC is a right triangle right angled at A
∠ BAC = 90°
Answer:
Given : In triangle ABC , AD is perpendicular to BC and AD² = BD.DC
To prove : BAC = 90°
Proof : in right triangles ∆ADB and ∆ADC
So, Pythagoras theorem should be apply ,
Then we have ,
AB² = AD² + BD² ----------(1)
AC²= AD²+ DC² ---------(2)
AB² + AC² = 2AD² + BD²+ DC²
= 2BD . CD + BD² + CD² [ ∵ given AD² = BD.CD ]
= (BD + CD )² = BC²
Thus in triangle ABC we have , AB² + AC²= BC²
hence triangle ABC is a right triangle right angled at A
∠ BAC = 90°