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In ΔABC, with the usual notations, if sin B sin C = bc/a2 , then the triangle is......
- a)Right angled triangle
- b)Obtuse angled triangle
- c)Equilateral triangle
- d)Acute angled triangle
Correct answer is option 'A'. Can you explain this answer?
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In ΔABC, with the usual notations, if sin B sin C = bc/a2 , then the ...
Key Idea: Use sine rule. I.e.
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a/Sin A = b/sinB = c/sin C
Given, sin B sin C = bc/a2
⇒ a2 = bc/sinBsinC
⇒ a2 = (b/sinB) (c/sinC)
⇒ a2 = (a/sinA)2
⇒ sin2 4 = 1
⇒ sinA = 1 sin90º
⇒ A = 90º
Most Upvoted Answer
In ΔABC, with the usual notations, if sin B sin C = bc/a2 , then the ...
Explanation:
Given: sin B sin C = bc/a^2
We know that in a right-angled triangle, the hypotenuse is the longest side and is opposite to the right angle. Let us assume that angle A is the right angle.
Proof:
Let us use the sine rule here:
sin B = b/cosec A
sin C = c/sin A
Substituting the values of sin B and sin C in the given equation, we get:
(b/cosec A) (c/sin A) = bc/a^2
Simplifying this, we get:
sin A = 1/2
This means that angle A is 30 degrees.
Now, we can use the Pythagorean theorem to find out if the triangle is indeed a right-angled triangle.
a^2 = b^2 + c^2
Substituting the value of sin A, we get:
a^2 = (bc/2)^2 + c^2
Simplifying this, we get:
a^2 = c^2 (1/4 + b^2/c^2)
Multiplying both sides by 4/c^2, we get:
(4a^2/c^2) = 1 + (4b^2/c^2)
Now, we know that sin B sin C = bc/a^2
Substituting the value of sin A, we get:
sin B sin C = (2bc/c^2)
Multiplying both sides by c^2/2, we get:
bc = sin B sin C (c^2/2)
Substituting this value in the equation we got earlier, we get:
(4a^2/c^2) = 1 + 2sin B sin C
Now, since sin B sin C = (2bc/c^2), we get:
(4a^2/c^2) = 1 + (4b^2/c^2)
This is the same equation we got earlier.
Therefore, the given condition is true only when the triangle is a right-angled triangle.
Given: sin B sin C = bc/a^2
We know that in a right-angled triangle, the hypotenuse is the longest side and is opposite to the right angle. Let us assume that angle A is the right angle.
Proof:
Let us use the sine rule here:
sin B = b/cosec A
sin C = c/sin A
Substituting the values of sin B and sin C in the given equation, we get:
(b/cosec A) (c/sin A) = bc/a^2
Simplifying this, we get:
sin A = 1/2
This means that angle A is 30 degrees.
Now, we can use the Pythagorean theorem to find out if the triangle is indeed a right-angled triangle.
a^2 = b^2 + c^2
Substituting the value of sin A, we get:
a^2 = (bc/2)^2 + c^2
Simplifying this, we get:
a^2 = c^2 (1/4 + b^2/c^2)
Multiplying both sides by 4/c^2, we get:
(4a^2/c^2) = 1 + (4b^2/c^2)
Now, we know that sin B sin C = bc/a^2
Substituting the value of sin A, we get:
sin B sin C = (2bc/c^2)
Multiplying both sides by c^2/2, we get:
bc = sin B sin C (c^2/2)
Substituting this value in the equation we got earlier, we get:
(4a^2/c^2) = 1 + 2sin B sin C
Now, since sin B sin C = (2bc/c^2), we get:
(4a^2/c^2) = 1 + (4b^2/c^2)
This is the same equation we got earlier.
Therefore, the given condition is true only when the triangle is a right-angled triangle.
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In ΔABC, with the usual notations, if sin B sin C = bc/a2 , then the triangle is......a)Right angled triangleb)Obtuse angled trianglec)Equilateral triangled)Acute angled triangleCorrect answer is option 'A'. Can you explain this answer?
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