Question

11,493 Learners found this answer helpful

find the smallest angle of δbcd.

Explanation:

Step1: Recall Triangle Angle-Side Relationship

In a triangle, the smallest angle is opposite the shortest side. First, identify the shortest side in \( \triangle BCD \). The sides are \( BC = 41 \, \text{m} \), \( BD = 34 \, \text{m} \), and \( CD = 45 \, \text{m} \). The shortest side is \( BD = 34 \, \text{m} \), so the angle opposite to it ( \( \angle C \)) is the smallest.

Step2: Apply the Law of Cosines

The Law of Cosines states \( c^2 = a^2 + b^2 - 2ab \cos(C) \), where \( a = 45 \), \( b = 41 \), and \( c = 34 \). Rearranging for \( \cos(C) \):
\( \cos(C) = \frac{a^2 + b^2 - c^2}{2ab} \)
Substitute values:
\( \cos(C) = \frac{45^2 + 41^2 - 34^2}{2 \times 45 \times 41} \)
Calculate numerator: \( 2025 + 1681 - 1156 = 2550 \)
Denominator: \( 2 \times 45 \times 41 = 3690 \)
\( \cos(C) = \frac{2550}{3690} \approx 0.6910 \)

Step3: Find Angle \( C \)

Take the arccosine: \( C = \arccos(0.6910) \approx 46.3^\circ \) (using a calculator).

Answer:

The smallest angle of \( \triangle BCD \) is approximately \( 46.3^\circ \) (opposite the side \( BD = 34 \, \text{m} \), i.e., \( \angle C \)).