the longest side of a triangle measures 2 and the shortest side measures 1. what cannot be the measurement of the angle between them?
a-30 degrees
b-60degrees
c-70 degrees
d-90degrees
I'd have to go with 90 degrees. That would make them the legs of a right triangle, but the hypotenuse is longer than either leg.
Or, analytically, using the law of cosines, let θ be the angle between the sides, and a be the length of the side opposite θ, and we get
a^2 = 1 + 4 - 4cosθ
cosθ = (5-a^2)/4
Now, we know that 1 < a < 2, so
1/4 < cosθ < 1
So, since cos 90 = 0, it does not fit the bill.
To determine the angle between the longest and shortest sides of a triangle, we can use the Law of Cosines. The formula states:
c² = a² + b² - 2ab * cos(C)
Where:
- c represents the length of the longest side (in this case, 2),
- a represents the length of the shortest side (in this case, 1),
- b represents the length of the other side (unknown in this case),
- C represents the angle between the sides with lengths c and a.
We can rearrange the formula to solve for the cosine of the angle:
cos(C) = (a² + b² - c²) / (2ab)
Now, let's substitute the known values into the formula and calculate the cosine of the angle:
cos(C) = (1² + b² - 2²) / (2 * 1 * b)
= (1 + b² - 4) / (2b)
= (b² - 3) / (2b)
Now, let's go through each option provided and check if the measurement of the angle (C) satisfies the formula:
a) 30 degrees:
cos(30 degrees) = (√3 / 2)
Substituting this value into the formula:
(√3 / 2) = (b² - 3) / (2b)
Simplifying this equation, we find that b = (√3 / 2) (approximately 0.866).
So, an angle of 30 degrees is possible.
b) 60 degrees:
cos(60 degrees) = 1/2
Substituting this value into the formula:
(1/2) = (b² - 3) / (2b)
Simplifying this equation, we find that b = (1/2) (approximately 0.5).
So, an angle of 60 degrees is possible.
c) 70 degrees:
cos(70 degrees) = approximately 0.342
Substituting this value into the formula:
0.342 = (b² - 3) / (2b)
Simplifying this equation, we find that b is approximately 0.805.
So, an angle of 70 degrees is possible.
d) 90 degrees:
cos(90 degrees) = 0
Substituting this value into the formula:
0 = (b² - 3) / (2b)
Simplifying this equation, we find that b = 0.
However, a side cannot have a length of 0, so an angle of 90 degrees is not possible.
Therefore, the measurement of the angle that cannot be the angle between the longest and shortest sides of the triangle is d) 90 degrees.