Triangle ABC is right angled with B is 90 D is the mid point of side BC . Prove that AC2=AD2+3CD2
Answers
Answer:
Sol:
D is the midpoint of BC. ⇒ AD = CD.
Angle B is a right angled triangle.
Consider ΔABC
AC2 = AB2 + BC2 [Pythagoras theorem]
⇒ AC2 = AB2 + (2BD)2
⇒ AC2 = AB2 + 4BD2 ----------- (1)
Consider ΔABC
AD2 = AB2 + BD2 [Pythagoras theorem] ----------- (2)
Subtracting equation (2) from (1), we get
⇒ AC2 - AD2 = 3BD2
⇒ AC2 - AD2 = 3CD2 [ Since BD = CD]
⇒ AC2 = AD2 + 3CD2
Hence proved.
Step-by-step explanation:
Given : Triangle ABC is right angled with B is 90 D is the mid point of side BC
To Find : Prove that AC²=AD²+3CD²
Solution:
Triangle ABC is right angled at B
Applying Pythagoras theorem
=> AC² = AB² + BC²
D is the mid point of side BC
=> BD = CD = BC/2
=> BC = 2CD
=> AC² = AB² + (2CD)²
=> AC² = AB² + 4CD²
=> AC² = AB² + CD² + 3CD²
CD = BD
=> AC² = AB² +BD² + 3CD²
AB² +BD² = AD²
=> AC² = AD² + 3CD²
QED
Hence Proved
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