Solve the triangle. $$ \mathrm{a}=24.8 \quad \mathrm{~b}=15.2 \quad \mathrm{c}=36.9 $$ What is the degree measure of angle A? $$ \square^{\circ} $$ (Simplify your answer. Type an integer or a decimal. Round to the nearest tenth if needed.) What is the degree measure of angle B ? $$ \square^{\circ} $$ (Simplify your answer. Type an integer or a decimal. Round to the nearest tenth if needed.) What is the degree measure of angle C ? $$ \square^{\circ} $$ (Simplify your answer. Type an integer or a decimal. Round to the nearest tenth if needed.)

Question

UpStudy AI · Accepted Answer

To solve the triangle with sides $$ a = 24.8 $$, $$ b = 15.2 $$, and $$ c = 36.9 $$, we can use the Law of Cosines to find the angles.

### Step 1: Find angle A

Using the Law of Cosines:
$$
\cos A = \frac{b^2 + c^2 - a^2}{2bc}
$$

Substituting the values:
$$
\cos A = \frac{15.2^2 + 36.9^2 - 24.8^2}{2 \cdot 15.2 \cdot 36.9}
$$

Calculating the squares:
$$
15.2^2 = 231.04, \quad 36.9^2 = 1369.61, \quad 24.8^2 = 615.04
$$

Now substituting these values:
$$
\cos A = \frac{231.04 + 1369.61 - 615.04}{2 \cdot 15.2 \cdot 36.9}
$$
$$
\cos A = \frac{985.61}{2 \cdot 15.2 \cdot 36.9}
$$
Calculating the denominator:
$$
2 \cdot 15.2 \cdot 36.9 = 1123.68
$$
Now substituting back:
$$
\cos A = \frac{985.61}{1123.68} \approx 0.876
$$

Now, finding angle A:
$$
A = \cos^{-1}(0.876) \approx 28.9^\circ
$$

### Step 2: Find angle B

Using the Law of Cosines again:
$$
\cos B = \frac{a^2 + c^2 - b^2}{2ac}
$$

Substituting the values:
$$
\cos B = \frac{24.8^2 + 36.9^2 - 15.2^2}{2 \cdot 24.8 \cdot 36.9}
$$

Using the previously calculated squares:
$$
\cos B = \frac{615.04 + 1369.61 - 231.04}{2 \cdot 24.8 \cdot 36.9}
$$
$$
\cos B = \frac{1753.61}{2 \cdot 24.8 \cdot 36.9}
$$
Calculating the denominator:
$$
2 \cdot 24.8 \cdot 36.9 = 1835.76
$$
Now substituting back:
$$
\cos B = \frac{1753.61}{1835.76} \approx 0.955
$$

Now, finding angle B:
$$
B = \cos^{-1}(0.955) \approx 16.2^\circ
$$

### Step 3: Find angle C

Using the fact that the sum of angles in a triangle is $$ 180^\circ $$:
$$
C = 180^\circ - A - B
$$
Substituting the values we found:
$$
C = 180^\circ - 28.9^\circ - 16.2^\circ \approx 134.9^\circ
$$

### Final Answers
- Angle A: $$ 28.9^\circ $$
- Angle B: $$ 16.2^\circ $$
- Angle C: $$ 134.9^\circ $$

Thus, the answers are:
- Angle A: $$ 28.9 $$
- Angle B: $$ 16.2 $$
- Angle C: $$ 134.9 $$
Angle A: $$ 28.9^\circ $$ Angle B: $$ 16.2^\circ $$ Angle C: $$ 134.9^\circ $$

Upstudy AI Solution

Answer

Solution

Step 1: Find angle A

Step 2: Find angle B

Step 3: Find angle C

Final Answers

Beyond the Answer

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