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If the angles of a triangle be in the ratio 1 : 4 : 5, then the ratio of the greatest side to the smallest side is
- a)1 : (√5−1)
- b)(√5+1) : 1
- c)5 : 1
- d)none of these
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
If the angles of a triangle be in the ratio 1 : 4 : 5, then the ratio ...
The sum of the angles of a triangle is 180 degrees. Let the angles be x, 4x, and 5x. Then we have:
x + 4x + 5x = 180
10x = 180
x = 18
So the angles are 18, 72, and 90 degrees. Now let's use the Law of Sines to find the ratios of the sides. For any triangle ABC, we have:
a/sin(A) = b/sin(B) = c/sin(C)
Let's use this formula for sides a and c, which are opposite angles x and 5x, respectively:
a/sin(x) = c/sin(5x)
Multiplying both sides by sin(x), we get:
a = c(sin(x)/sin(5x))
Now let's use this formula for sides b and a, which are opposite angles 4x and x, respectively:
b/sin(4x) = a/sin(x)
Multiplying both sides by sin(4x), we get:
b = a(sin(4x)/sin(x))
Now we can use these equations to find the ratio of the greatest side to the smallest side:
greatest side/smallest side = c/a
Substituting the expressions for c and a from above, we get:
greatest side/smallest side = (sin(x)/sin(5x))/(sin(4x)/sin(x))
Simplifying this expression, we get:
greatest side/smallest side = (sin(x)^2)/(sin(4x)sin(5x))
Now we just need to plug in x = 18 degrees and simplify:
greatest side/smallest side = (sin(18)^2)/(sin(4(18))sin(5(18)))
greatest side/smallest side = (0.155)^2/(0.809)(0.951)
greatest side/smallest side = 0.0195/0.771
greatest side/smallest side = 0.0253
Therefore, the ratio of the greatest side to the smallest side is approximately 0.0253, or 1:39.5 (rounded to one decimal place).
x + 4x + 5x = 180
10x = 180
x = 18
So the angles are 18, 72, and 90 degrees. Now let's use the Law of Sines to find the ratios of the sides. For any triangle ABC, we have:
a/sin(A) = b/sin(B) = c/sin(C)
Let's use this formula for sides a and c, which are opposite angles x and 5x, respectively:
a/sin(x) = c/sin(5x)
Multiplying both sides by sin(x), we get:
a = c(sin(x)/sin(5x))
Now let's use this formula for sides b and a, which are opposite angles 4x and x, respectively:
b/sin(4x) = a/sin(x)
Multiplying both sides by sin(4x), we get:
b = a(sin(4x)/sin(x))
Now we can use these equations to find the ratio of the greatest side to the smallest side:
greatest side/smallest side = c/a
Substituting the expressions for c and a from above, we get:
greatest side/smallest side = (sin(x)/sin(5x))/(sin(4x)/sin(x))
Simplifying this expression, we get:
greatest side/smallest side = (sin(x)^2)/(sin(4x)sin(5x))
Now we just need to plug in x = 18 degrees and simplify:
greatest side/smallest side = (sin(18)^2)/(sin(4(18))sin(5(18)))
greatest side/smallest side = (0.155)^2/(0.809)(0.951)
greatest side/smallest side = 0.0195/0.771
greatest side/smallest side = 0.0253
Therefore, the ratio of the greatest side to the smallest side is approximately 0.0253, or 1:39.5 (rounded to one decimal place).
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If the angles of a triangle be in the ratio 1 : 4 : 5, then the ratio of the greatest side to the smallest side isa)1 : (√5−1)b)(√5+1) : 1c)5:1d)none of theseCorrect answer is option 'B'. Can you explain this answer?
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