Labeling the angles of a triangle as A, B, and C and the lengths of the corresponding opposite sides as a, b and c. Given b=7 and A=30°. If the triangle is identified and unique, the value of a satisfies.
A) a<3.5 B) a>3.5 C) 3.5<a<8 D) a>8 E) none
Draw altitude from C to side c
sin 30 = its length /7
but sin 30 = 1/2
so it is 3.5
EVERY other line from C to c is longer than that
so
a > 3.5
If a > 3.5 then ABC is not unique, as there are two solutions, one with C obtuse (a > 7.5) and the other with B obtuse (a < 3.5)
The only unique solution I see is where a = 3.5
Oops - typo. I should have said
two solutions, one with C obtuse (c > 7cosA) and the other with B obtuse (c < 7cosA)
To solve this problem, we can make use of the Law of Sines, which states that the ratio of the length of a side of a triangle to the sine of the opposite angle is constant.
The Law of Sines can be written as:
a/sinA = b/sinB = c/sinC
Given b=7 and A=30°, we can plug in these values into the Law of Sines to find the value of a.
a/sin(30°) = 7/sinB
To isolate a, we can cross-multiply and divide by sin(30°):
a = (7*sin(30°)) / sinB
Now, let's analyze the possible values for sinB:
Since angles in a triangle add up to 180°, we can write:
B + C = 180°
Since the given angle A = 30°, we can also write:
A + B + C = 180°
30° + B + C = 180°
B + C = 150°
Knowing that B + C = 150°, we can now find the range of possible values for sinB.
The maximum value of sinB occurs when B = 90° and C = 60°:
sinB = sin(90°) = 1
The minimum value of sinB occurs when B = 0° and C = 150°:
sinB = sin(0°) = 0
Since sinB can vary between 0 and 1, we can find the range of possible values for a by plugging in these extreme values into our equation for a:
a = (7*sin(30°)) / 1 (when sinB = 1)
a = 7*sin(30°) = 7 * (1/2) = 3.5
a = (7*sin(30°)) / 0 (when sinB = 0)
Here, we have an undefined value since we cannot divide by zero.
Since the range of possible values for a is greater than 3.5 (3.5 < a < ∞), we can conclude that the correct answer is C) 3.5 < a < 8.