plz help me in solving this question. In the figure, ∠BAD = 90°. (1) Given that AB= 36cm and AD= 48cm, find the length of BD. (2) Given further that BC= 87cm and CD= 63cm , show that ΔBCD is a right-angled triangle. plz give me detailed and step by step answer.
Answers
Step-by-step explanation:
here is the answer and the step by step explanation.
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Answer:
1) The length of BD is 36 cm.
2) ΔBDC is a right-angled triangle.
Step-by-step-explanation:
NOTE: Refer to the attachment for the diagram.
1)
In figure, in ΔBAD, m∠BAD = 90°. - - [ Given ]
∴ ΔBAD is a right-angled triangle.
∴ ( BD )² = ( AB )² + ( AD )² - - [ Pythagors theorem ]
→ ( BD )² = ( 36 )² + ( 48 )² - - [ Given ]
→ ( BD )² = 1296 + 2304
→ ( BD )² = 3600
→ BD = √(3600)
→ BD = √( 60 × 60 )
∴ BD = 60 cm - - ( 1 )
2)
In figure, in ΔBCD,
BC = 87 cm
- - - } Given
CD = 63 cm
We have the value of BD from equation ( 1 ).
∴ BD = 60 cm
Now,
The longest side in ΔBCD is BC.
∴ ( BC )² = ( 87 )²
→ ( BC )² = 87 × 87
→ ( BC )² = 7569 - - ( 2 )
Now,
The sum of squares of ramaining two sides is
( BD )² + ( CD )² = ( 60 )² + ( 63 )²
→ ( BD )² + ( CD )² = 60 × 60 + 63 × 63
→ ( BD )² + ( CD )² = 3600 + 3969
→ ( BD )² + ( CD )² = 7569 - - ( 3 )
From equation ( 2 ) and ( 3 ),
( BC )² = ( BD )² + ( CD )²
Hence,
The square of the longest side in a triangles is equal to the sum of the squares of the remaining two sides.
∴ By converse of Pythagors theorem,
ΔBDC is a right-angled triangle.
Additional Information:
1. Pythagors theorem:
1. This is a theorem related to right-angled triangle.
2. This theorem was given by Mathematician Pythagors.
3. This theorem says that,
In a right-angled triangle, the square of the longest side is equal to the sum of the squares of the remaining two sides.
4. The longest side of a right-angled triangle is also called as hypotenuse.
2. Converse of Pythagors theorem:
If the square of the longest side of a triangle is equal to the sum of the squares of the remaining two sides, then the triangle is a right-angled triangle.
