Question

Math

Posted 2 months ago

```
The following figure shows $\triangle A B C$ with side lengths to the nearest tenth.
Find $A C$ in $\triangle A B C$.
Round to the nearest tenth.
$A C=$
$\square$
```

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Answer from Sia

Posted 2 months ago

Solution by Steps

step 1

To find the length of side AC, we can use the Law of Sines, which relates the lengths of sides of a triangle to the sines of its opposite angles

step 2

The Law of Sines states that $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$, where $a$, $b$, and $c$ are the lengths of the sides opposite to angles $A$, $B$, and $C$ respectively

step 3

Given that $AB = 12$ units, $\angle C = 40^\circ$, and $\angle B = 110^\circ$, we can find $\angle A$ by using the fact that the sum of angles in a triangle is $180^\circ$

step 4

Calculating $\angle A$: $\angle A = 180^\circ - \angle B - \angle C = 180^\circ - 110^\circ - 40^\circ = 30^\circ$

step 5

Now we can apply the Law of Sines to find $AC$: $\frac{AC}{\sin B} = \frac{AB}{\sin A}$

step 6

Substituting the known values into the Law of Sines: $\frac{AC}{\sin 110^\circ} = \frac{12}{\sin 30^\circ}$

step 7

Solving for $AC$: $AC = \frac{12 \cdot \sin 110^\circ}{\sin 30^\circ}$

step 8

Since $\sin 30^\circ = \frac{1}{2}$, the equation simplifies to: $AC = \frac{12 \cdot \sin 110^\circ}{\frac{1}{2}} = 24 \cdot \sin 110^\circ$

step 9

Calculate $AC$ using a calculator and round to the nearest tenth: $AC \approx 24 \cdot \sin 110^\circ$

Answer

[Insert final answer here after calculation]

Key Concept

Law of Sines

Explanation

The Law of Sines is used to find unknown side lengths in a triangle when given two angles and one side.

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