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The vertices angle of a triangle is divided into two parts, such that the tangent of one part is 3 times the tangent of the other and the difference of these parts is 30º, then the triangle is
- a)Isosceles
- b)Right angled
- c)Obtuse angled
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
The vertices angle of a triangle is divided into two parts, such that ...
Let's assume that the two parts of the triangle's angles are x and y, where x > y.
According to the given conditions, we can write the following equations:
tan(x) = 3 * tan(y) ...(1)
x - y = 30 ...(2)
From equation (2), we can express x in terms of y:
x = y + 30 ...(3)
Now, substitute equation (3) into equation (1):
tan(y + 30) = 3 * tan(y)
Using the tangent addition formula, we have:
(tan(y) + tan(30))/(1 - tan(y) * tan(30)) = 3 * tan(y)
Expanding this equation, we get:
tan(y) + sqrt(3)/3 = 3 * tan(y) - sqrt(3) * tan(y)
Simplifying further:
sqrt(3)/3 = tan(y) * (3 - 1 - sqrt(3))
sqrt(3)/3 = tan(y) * (2 - sqrt(3))
Dividing both sides by (2 - sqrt(3)), we get:
tan(y) = sqrt(3)/3 * (1/(2 - sqrt(3)))
Rationalizing the denominator, we have:
tan(y) = sqrt(3)/3 * (2 + sqrt(3))/(4 - 3)
tan(y) = sqrt(3)(2 + sqrt(3))/3
Now, we can find the value of y by taking the inverse tangent (arctan) of both sides:
y = arctan(sqrt(3)(2 + sqrt(3))/3)
Using a calculator, we find that y ≈ 61.7 degrees.
To find the value of x, we can substitute this value of y into equation (3):
x = y + 30
x = 61.7 + 30
x ≈ 91.7 degrees.
Therefore, the two parts of the triangle's angles are approximately 61.7 degrees and 91.7 degrees.
According to the given conditions, we can write the following equations:
tan(x) = 3 * tan(y) ...(1)
x - y = 30 ...(2)
From equation (2), we can express x in terms of y:
x = y + 30 ...(3)
Now, substitute equation (3) into equation (1):
tan(y + 30) = 3 * tan(y)
Using the tangent addition formula, we have:
(tan(y) + tan(30))/(1 - tan(y) * tan(30)) = 3 * tan(y)
Expanding this equation, we get:
tan(y) + sqrt(3)/3 = 3 * tan(y) - sqrt(3) * tan(y)
Simplifying further:
sqrt(3)/3 = tan(y) * (3 - 1 - sqrt(3))
sqrt(3)/3 = tan(y) * (2 - sqrt(3))
Dividing both sides by (2 - sqrt(3)), we get:
tan(y) = sqrt(3)/3 * (1/(2 - sqrt(3)))
Rationalizing the denominator, we have:
tan(y) = sqrt(3)/3 * (2 + sqrt(3))/(4 - 3)
tan(y) = sqrt(3)(2 + sqrt(3))/3
Now, we can find the value of y by taking the inverse tangent (arctan) of both sides:
y = arctan(sqrt(3)(2 + sqrt(3))/3)
Using a calculator, we find that y ≈ 61.7 degrees.
To find the value of x, we can substitute this value of y into equation (3):
x = y + 30
x = 61.7 + 30
x ≈ 91.7 degrees.
Therefore, the two parts of the triangle's angles are approximately 61.7 degrees and 91.7 degrees.
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The vertices angle of a triangle is divided into two parts, such that the tangent of one part is 3 times the tangent of the other and the difference of these parts is 30º, then the triangle isa)Isoscelesb)Right angledc)Obtuse angledd)None of theseCorrect answer is option 'B'. Can you explain this answer?
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