Final Answer:
The length of the shorter base is approximately 7.23 cm.
Step-by-step explanation:
Identify relevant information:
Longer base length (BL) = 12 cm
Side length (S) = 5 cm
Acute angle (θ) = 37°
Formulate a relationship between the sides:
We can utilize the Pythagorean theorem on the right triangle formed by the longer base, the side, and the hypotenuse (which is also the longer base in an isosceles trapezoid):
BL^2 = 2 * S^2
Substitute known values and solve for S:
12^2 = 2 * 5^2
S^2 = 12^2 / 2
S ≈ 7.23 cm
Calculate the shorter base length:
Since the base angles are congruent, we can apply the sine rule on the smaller right triangle formed by the shorter base, the side, and the acute angle:
S / sin(θ) = BL / sin(90° - θ)
Substitute and solve for the shorter base:
7.23 / sin(37°) = 12 / sin(90° - 37°)
Shorter Base ≈ 7.23 cm