Question

find the smallest angle of the triangle 2x 96 x+12

Answer 1

To find the smallest angle of the triangle with sides \(2x\), \(96\), and \(x + 12\), we first need to determine the lengths of the sides based on the value of \(x\).

1. **Identify the Side Lengths**:
- Side 1: \(2x\)
- Side 2: \(96\)
- Side 3: \(x + 12\)

2. **Determine Constraints for a Triangle**: We must ensure that the triangle inequality holds for the three sides:
- \(2x + (x + 12) > 96\)
- \(2x + 96 > (x + 12)\)
- \((x + 12) + 96 > 2x\)

Let's simplify these inequalities one by one.

- **First Inequality**:
\[
2x + (x + 12) > 96 \implies 3x + 12 > 96 \implies 3x > 84 \implies x > 28
\]

- **Second Inequality**:
\[
2x + 96 > (x + 12) \implies 2x + 96 > x + 12 \implies x > -84 \quad \text{(This inequality is always true for } x > 28\text{)}
\]

- **Third Inequality**:
\[
(x + 12) + 96 > 2x \implies x + 108 > 2x \implies 108 > x \implies x < 108
\]

So, we have the conditions:
\[
28 < x < 108
\]

3. **Finding Side Lengths**:
Choose a value of \(x\) within the interval \(28 < x < 108\). Let's use \(x = 30\) as an example:
- Side 1: \(2x = 2(30) = 60\)
- Side 2: \(96\)
- Side 3: \(x + 12 = 30 + 12 = 42\)

The side lengths are \(60\), \(96\), and \(42\).

4. **Determine the Smallest Angle**:
The smallest angle is opposite the shortest side. Here:
- Shortest side = \(42\) (opposite angle we'll denote as \(A\)).
- The angles will be calculated using the Law of Cosines.

However, since we have three sides, we can find angles \(A\), \(B\), and \(C\) using:
- \(A\): opposite to \(42\)
- \(B\): opposite to \(60\)
- \(C\): opposite to \(96\)

The Law of Cosines states:

\[
A = \cos^{-1}\left(\frac{b^2 + c^2 - a^2}{2bc}\right)
\]
where \(a\), \(b\), and \(c\) are the lengths of the sides of the triangle.

Plugging in \(A\)'s values:
\[
A = \cos^{-1}\left(\frac{60^2 + 96^2 - 42^2}{2 \cdot 60 \cdot 96}\right)
\]

Simplifying:
- \(60^2 = 3600\)
- \(96^2 = 9216\)
- \(42^2 = 1764\)

Now, plug these values into the equation:
\[
A = \cos^{-1}\left(\frac{3600 + 9216 - 1764}{2 \cdot 60 \cdot 96}\right)
\]
\[
A = \cos^{-1}\left(\frac{11052}{11520}\right)
\]
\[
A = \cos^{-1}\left(0.96\right)
\]

Using a calculator, you can find this angle and, ultimately, confirm that it is indeed the smallest angle (as it corresponds to the smallest side).

**Conclusion**:
You can compute \(A\) for any value of \(x\) within the defined range (\(28 < x < 108\)), but typically the smallest angle occurs opposite the smallest side. Thus, using the Law of Cosines will show that the angle opposite the side \(42\) is the smallest angle in the triangle given \(x\).