Solve the triangle given to the right.
27
18
14
A
B
C
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Part 1
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To solve this triangle, where we are given the lengths of all three sides (27, 18, and 14), we can use the **Law of Cosines** to find each angle.

The Law of Cosines states:
\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]
where \( c \) is the side opposite angle \( C \), and similarly for other sides and angles.

Let’s label the triangle as follows:
- Side \( a = 27 \), opposite angle \( A \)
- Side \( b = 18 \), opposite angle \( B \)
- Side \( c = 14 \), opposite angle \( C \)

We’ll calculate each angle:

### Step 1: Find \( \angle A \) using the Law of Cosines
\[
\cos(A) = \frac{b^2 + c^2 - a^2}{2bc}
\]
Substitute the values:
\[
\cos(A) = \frac{18^2 + 14^2 - 27^2}{2 \cdot 18 \cdot 14}
\]

### Step 2: Find \( \angle B \) using the Law of Cosines
\[
\cos(B) = \frac{a^2 + c^2 - b^2}{2ac}
\]
Substitute the values:
\[
\cos(B) = \frac{27^2 + 14^2 - 18^2}{2 \cdot 27 \cdot 14}
\]

### Step 3: Find \( \angle C \) using the Law of Cosines
\[
\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}
\]
Substitute the values:
\[
\cos(C) = \frac{27^2 + 18^2 - 14^2}{2 \cdot 27 \cdot 18}
\]

Once we calculate these values, we can find \( \angle A \), \( \angle B \), and \( \angle C \) by taking the arccosine (inverse cosine) of each result. Let me calculate these for you.The angles of the triangle are approximately:

- \( A \approx 114.5^\circ \)
- \( B \approx 37.3^\circ \)
- \( C \approx 28.2^\circ \)

Let me know if you’d like further details on the calculations.

### Related Questions
1. How is the Law of Cosines derived?
2. How can we confirm that the angles add up to 180 degrees?
3. Can the Law of Sines be used with this triangle as well?
4. What is the difference between solving a triangle given three sides vs. two sides and an angle?
5. How do we solve for missing angles in a right triangle?

**Tip:** When using the Law of Cosines, ensure you substitute the correct values for each side to avoid calculation errors.

Solve the triangle given to the right. 27, 18, 14, A, B, C

Solve a triangle with sides of 27, 18, and 14 using the Law of Cosines. Find angles A, B, and C by applying trigonometric formulas to determine each angle's measure. Suitable for students in Grades 10-12.

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