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State True or False.
In the given figure, AD is a median of a triangle ABC and AE ⊥ BC. Then

$A C 2 + A B 2 = 2 A D 2 + 12 B C 2$

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True

False

answer is A.

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Detailed Solution

Given that in the figure below, AD is a median of a triangle ABC and AE ⊥ BC.
Question Image

Since AD is median,

B D = D C = 12 BC

We know Pythagoras theorem states that in a right-angled triangle,

(h y p o t e n u s e) 2 = (b a s e) 2 + (p e r p e n d i c u l a r) 2

.
In

Δ A E D

, using Pythagoras theorem,

A D 2 = A E 2 + D E 2

⇒ A E 2 = A D 2 − D E 2

………(1)
In

Δ A E B

, using Pythagoras theorem,
Question Image

Now in

Δ A E C

,using Pythagoras theorem
Question Image

Adding [2] and [3], we get

⇒ AB 2 + AC 2 = BD 2 + AD 2 + 2 ⋅ BD ⋅ DE + AD 2 + BD 2 − 2 ⋅ BD ⋅ DE ⇒ AB 2 + AC 2 = 2 A D 2 + 2 B D 2 ⇒ AB 2 + AC 2 = 2 A D 2 + 2 ⋅ B C 2 4 B D = ½ B C ⇒ AB 2 + AC 2 = 2 A D 2 + B C 2 2

So, the given statement is true.
Therefore the correct option is 1.

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