Questions: ABC is a triangle with angle A=30 degrees, angle B=135 degrees, and BC=8 cm. Find AC.

Transcript text: ABC is a triangle with $\angle A=30^{\circ}, \angle B=135^{\circ}$, and $\mathrm{BC}=8 \mathrm{~cm}$. Find AC.

Solution

Solution Steps

To find the length of side AC in triangle ABC, we can use the Law of Sines. Given the angles and one side, we can set up the ratio of the sides to the sines of their opposite angles.

Solution Approach

Use the fact that the sum of angles in a triangle is 180° to find the third angle, ∠C.
Apply the Law of Sines to find the length of AC.

Step 1: Calculate Angle C

In triangle ABC, we know the angles $ \angle A = 30^\circ $ and $ \angle B = 135^\circ $. To find the third angle $ \angle C $, we use the fact that the sum of angles in a triangle is $ 180^\circ $:

\[ \angle C = 180^\circ - \angle A - \angle B = 180^\circ - 30^\circ - 135^\circ = 15^\circ \]

Step 2: Apply the Law of Sines

Using the Law of Sines, we can relate the sides and angles of the triangle:

\[ \frac{AC}{\sin(B)} = \frac{BC}{\sin(C)} \]

Substituting the known values:

\[ \frac{AC}{\sin(135^\circ)} = \frac{8}{\sin(15^\circ)} \]

Step 3: Calculate Sine Values

We calculate the sine values:

\[ \sin(135^\circ) = \frac{\sqrt{2}}{2} \quad \text{and} \quad \sin(15^\circ) \approx 0.2588 \]

Step 4: Solve for AC

Now we can solve for $ AC $:

\[ AC = \frac{8 \cdot \sin(135^\circ)}{\sin(15^\circ)} = \frac{8 \cdot \frac{\sqrt{2}}{2}}{0.2588} \approx 21.8564 \]