Question: Help: PDF Ex. 18.5, Video Ex 18.5
12. Find the missing sides. Don’t use a calculator, use reference triangles.
Select one:
A = 8, B = 2\sqrt{3}
A = 4, B = 2\sqrt{3}
A = 8, B = 4\sqrt{3}
A = 4, B = 4\sqrt{3}
Graph shape: a right triangle is shown at the top-right; the left vertical leg is labeled B, the slanted right side is labeled A, the bottom horizontal side is labeled 4, the top angle is 30°, and the right angle is at the bottom-left.
1. **Problem statement:** We have a right triangle with a right angle at the bottom-left, a 30° angle at the top, the bottom horizontal side length is $4$, the vertical leg is $B$, and the hypotenuse is $A$. We need to find the missing sides $A$ and $B$ without using a calculator, using reference triangles.
2. **Recall the 30°-60°-90° triangle ratios:** In a 30°-60°-90° triangle, the sides are in the ratio:
$$1 : \sqrt{3} : 2$$
where the side opposite 30° is $1$, opposite 60° is $\sqrt{3}$, and the hypotenuse is $2$.
3. **Identify the sides relative to the 30° angle:**
- The side opposite 30° is the vertical leg $B$.
- The side adjacent to 30° (bottom side) is $4$.
- The hypotenuse is $A$.
4. **Match the known side to the ratio:**
The bottom side corresponds to the side opposite 60°, which is $\sqrt{3}$ in the ratio.
Let the scale factor be $k$.
Then:
$$4 = k \times \sqrt{3} \implies k = \frac{4}{\sqrt{3}}$$
5. **Find $B$ (opposite 30°):**
$$B = k \times 1 = \frac{4}{\sqrt{3}}$$
6. **Find $A$ (hypotenuse):**
$$A = k \times 2 = 2 \times \frac{4}{\sqrt{3}} = \frac{8}{\sqrt{3}}$$
7. **Rationalize denominators:**
$$B = \frac{4}{\sqrt{3}} = \frac{4 \sqrt{3}}{3}$$
$$A = \frac{8}{\sqrt{3}} = \frac{8 \sqrt{3}}{3}$$
8. **Compare with options:**
Options have $B$ as multiples of $\sqrt{3}$ and $A$ as integers.
Multiply numerator and denominator by $\sqrt{3}$ to get approximate values:
- $B = \frac{4 \sqrt{3}}{3} \approx 2.31$
- $A = \frac{8 \sqrt{3}}{3} \approx 4.62$
Since the bottom side is $4$, the hypotenuse $A$ should be larger than $4$, and $B$ should be less than $4$.
9. **Check options:**
- $A=8, B=2\sqrt{3} \approx 3.46$ (too large for $A$)
- $A=4, B=2\sqrt{3} \approx 3.46$ (hypotenuse cannot be $4$ if bottom side is $4$)
- $A=8, B=4\sqrt{3} \approx 6.93$ (too large for $B$)
- $A=4, B=4\sqrt{3} \approx 6.93$ (hypotenuse cannot be $4$)
10. **Re-examine the triangle labeling:** The bottom side is $4$, adjacent to 30°, so it corresponds to $\sqrt{3}k$ side.
If bottom side is $4$, then $k = \frac{4}{\sqrt{3}}$.
Then $B = k \times 1 = \frac{4}{\sqrt{3}} = \frac{4 \sqrt{3}}{3}$, which is approximately $2.31$.
None of the options exactly match $\frac{4 \sqrt{3}}{3}$, but $2\sqrt{3} = 2 \times 1.732 = 3.464$ is closest.
11. **Alternative approach:** Maybe the bottom side corresponds to the side opposite 30°, so $4 = k \times 1$, then $k=4$.
Then:
$$B = k \times \sqrt{3} = 4 \sqrt{3}$$
$$A = k \times 2 = 8$$
This matches option:
$$A=8, B=4\sqrt{3}$$
12. **Conclusion:** The bottom side is opposite 30°, so $4 = k \times 1$, giving $k=4$, then $B=4\sqrt{3}$ and $A=8$.
**Final answer:**
$$\boxed{A=8, B=4\sqrt{3}}$$