Triangle ABC has an altitude BN which is 6 cm long. AN=9 cm and NC=4 cm. Show whether ABC is a right angled triangle. (Show all the working).
Answers
EXPLANATION.
Triangle ABC has an altitude BN which is 6 cm long.
⇒ AN = 9cm.
⇒ NC = 4cm.
As we know that,
Pythagoras theorem.
⇒ H² = P² + B².
Hypotenuse > Perpendicular > Base.
In right angled triangle (ΔBNC).
⇒ (BC)² = (BN)² + (NC)².
⇒ (BC)² = (6)² + (4)².
⇒ (BC)² = 36 + 16.
⇒ (BC)² = 52cm.
In right angled triangle (ΔBNA).
⇒ (BA)² = (BN)² + (NA)².
⇒ (BA)² = (6)² + (9)².
⇒ (BA)² = 36 + 81.
⇒ (BA)² = 117cm.
According to Pythagoras theorem.
⇒ (AC)² = (AB)² + (BC)².
⇒ AC = AN + NC.
⇒ AC = 9 + 4.
⇒ AC = 13cm.
Put the value in the equation, we get.
⇒ (13)² = 117 + 52.
⇒ 169 = 169.
Hence Proved.
Given :-
Triangle ABC has an altitude BN which is 6 cm long
To Proof :-
Show whether ABC is a right angled triangle. (Show all the working).
Solution :-
According to the Pythagoras theorem
H² = P² + B²
Here
H = BC
P = BN
B = NC
Now
BC² = 4² + 6²
BC² = 16 + 36
BC² = 52
BC = √52 cm
Now
H = BA
B = NA
P = BN
BA² = 6² + 9²
BA² = 36 + 81
BA² = 117
BA = √117 cm
So,
Length of AC = Length of AN + Length of NC
Length of AC = 9 + 4
Length of AC = 13 cm
By using pythagoras theorem
(13)² = (√117)² + (√52)²
169 = 117 + 52
169 = 169
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