We used the sine function and the Pythagorean theorem to find that a is 7/2 and b is 7√3/2 . Option B is the correct option.
Step 1: Identify the trigonometric relationship
We see a 30-60-90 triangle where side b is the hypotenuse, side a is half the hypotenuse, and the angle at A is 30 degrees. We can use the trigonometric ratio sine (sin) to relate these sides:
sin(30°) = a / b
Step 2: Substitute and solve for a
We know the value of sin(30°) is 1/2. Substituting this :
1/2 = a / 7
a = 1/2 * 7
a = 7/2
Step 3: Calculate b using the Pythagorean theorem
We can also use the Pythagorean theorem to find b, although it’s already given in the question.
a^2 + b^2 = c^2
Substituting a = 7/2 and c = 7 (the hypotenuse):
(7/2)^2 + b^2 = 7^2
49/4 + b^2 = 49
b^2 = 49 - 49/4
b^2 = 147/4
Taking the square root of both sides:
b = (7√3)/2
Therefore, the values of a and b in the triangle are:
a = 7/2
b = (7√3)/2