Answer :
The proof is as follows :
Step-by-step explanation:
Let NC = x
⇒ AB = 3x and AN = 2x
In Δ ABN, By using Pythagoras theorem,
AB² = BN² + AN²
⇒ BN² = AB² - AN²
⇒ BN² = (3x)² - (2x)²
⇒ BN² = 5x²
⇒ BN = x√5 .......................(1)
Now in ΔANC , Using Pythagoras theorem We have,
AC² = NC² + AN²
⇒ AC² = x² + (2x)²
⇒ AC² = 5x²
⇒ AC = x√5 ....................(2)
From equations (1) and (2) We get,
AC = BN , which is our required result
Answer:
BN=AC=√5 x.
The proof is explained in step-by-step explaination.
Step-by-step explanation:
Let NC=x. It is given that AB=3NC & AN=2NC
⇒ AB=3x & AN=2x
By applying Pythagoras theorem
In triangle ANC,
[tex]AC^{2}=AN^{2}+NC^{2}[/tex]
⇒ [tex]AC^{2} = (2x)^{2}+x^{2}[/tex]
⇒ [tex]AC^{2}=4x^{2}+x^{2} =5x^{2}[/tex]
⇒ [tex]AC=\sqrt{5}x[/tex] → (1)
Similarly, In triangle ABN,
[tex]AB^{2}=AN^{2}+BN^{2}[/tex]
⇒ [tex](3x)^{2}=BN^{2}+x^{2}[/tex]
⇒ [tex]9x^{2} = (BN)^{2}+4x^{2}[/tex]
⇒ [tex]BN^{2}=5x^{2}[/tex]
⇒ [tex]BN=\sqrt{5}x[/tex] → (2)
From eq (1) & (2), AC=BN
