Given: angle A is 20˚, b = 50 cm, for what possible range of lengths of side “a” would triangle ABC have two possible solutions (approximate to 2 decimal places)?
To determine the possible range of lengths for side "a" in triangle ABC, we need to use the Law of Sines.
The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. Mathematically, it can be written as:
a/sin(A) = b/sin(B) = c/sin(C)
In this case, we are given angle A as 20˚ and side b as 50 cm. We are interested in finding the range of lengths for side "a" that would result in two possible solutions.
When dealing with triangles, there are a few rules to consider:
1) The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This is known as the Triangle Inequality Theorem.
2) An angle cannot be greater than 180 degrees in a triangle.
Now, let's use these concepts to solve the problem:
1) Determine angle B: Since angle C is 180˚ - 20˚ - 90˚ = 70˚ (the sum of angles in a triangle is 180˚), angle B can be found using the Triangle Inequality Theorem:
angle B = 180˚ - 20˚ - 70˚ = 90˚
2) Use the Law of Sines to find the length of side "a":
a/sin(20˚) = 50/sin(90˚)
Rearrange the equation to solve for "a":
a = 50 * sin(20˚)/sin(90˚)
Calculate "a":
a ≈ 29.23 cm
Now, we need to consider the range of lengths for side "a" that would result in two possible solutions.
Since angle A is 20˚, we can see that side "a" is the shortest side in the triangle. Side "a" needs to be longer than the difference between side b and side c to have two possible solutions.
Let's assume side a = 29.23 cm (the value we calculated earlier). In this case, we know that side a is the smallest side, and the difference between side b and c will be the smallest. Let's call this difference "d".
d = |b - c|
d = |50 - c|
We can set up an inequality:
a > d
29.23 cm > |50 - c|
Now, we can solve this inequality to find the possible range for side "a":
1) c < 50 - 29.23
c < 20.77 cm
2) c > 50 + 29.23
c > 79.23 cm
Therefore, the possible range of lengths for side "a" in triangle ABC, to have two possible solutions, is a > 20.77 cm and a < 79.23 cm (approximate to 2 decimal places).