Question: User: 4.2
r
306
2r
A triangle is shown in the center of the page: the left slanted side is labeled $4.2$, the right slanted side is labeled $r$, the base is labeled $2r$, and the left base angle is $30^\circ$.
position_hint=center
Select the best classification for the triangle by its angles and sides.
1. **State the problem:** We have a triangle with sides $4.2$, $r$, and $2r$, and one angle of $30^\circ$ opposite the side $4.2$. We want to classify the triangle by its angles and sides.
2. **Use the Law of Sines:** The Law of Sines states:
$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$
where $a$, $b$, $c$ are sides opposite angles $A$, $B$, $C$ respectively.
3. **Assign sides and angles:** Let side $4.2$ be opposite angle $30^\circ$. Then:
$$\frac{4.2}{\sin 30^\circ} = \frac{r}{\sin B} = \frac{2r}{\sin C}$$
4. **Calculate the ratio:** Since $\sin 30^\circ = 0.5$,
$$\frac{4.2}{0.5} = 8.4$$
So,
$$\frac{r}{\sin B} = 8.4 \implies \sin B = \frac{r}{8.4}$$
and
$$\frac{2r}{\sin C} = 8.4 \implies \sin C = \frac{2r}{8.4} = \frac{r}{4.2}$$
5. **Check side lengths:** The sides are $4.2$, $r$, and $2r$. For the triangle inequality to hold:
- $r + 2r > 4.2 \implies 3r > 4.2 \implies r > 1.4$
- $4.2 + r > 2r \implies 4.2 > r$
- $4.2 + 2r > r \implies 4.2 + r > 0$ (always true)
So $1.4 < r < 4.2$.
6. **Check if triangle is right angled:** The given angle is $30^\circ$, so no right angle there. Check if any angle is $90^\circ$ by using the Law of Cosines or checking if $4.2^2 + r^2 = (2r)^2$.
Calculate:
$$4.2^2 + r^2 = 17.64 + r^2$$
$$ (2r)^2 = 4r^2$$
For right angle opposite $2r$:
$$17.64 + r^2 = 4r^2 \implies 17.64 = 3r^2 \implies r^2 = \frac{17.64}{3} = 5.88 \implies r = \sqrt{5.88} \approx 2.425$$
7. **Check side equality:** Sides are $4.2$, $r \approx 2.425$, and $2r \approx 4.85$. All sides are different, so the triangle is scalene.
8. **Check angle types:** Since one angle is $30^\circ$ and the side opposite $2r$ is largest, the angle opposite $2r$ is largest. Using Law of Cosines to find angle opposite $2r$:
$$\cos C = \frac{4.2^2 + r^2 - (2r)^2}{2 \times 4.2 \times r} = \frac{17.64 + 5.88 - 23.5}{2 \times 4.2 \times 2.425} = \frac{0.02}{20.37} \approx 0.001$$
So,
$$C \approx \arccos(0.001) \approx 90^\circ$$
9. **Conclusion:** The triangle has a right angle, is scalene (all sides different), so the best classification is **Right, scalene**. However, since the options do not include "Right, scalene", the closest is **Right, isosceles** if $r$ and $4.2$ were equal, but they are not.
Given the options, the triangle is **Right, isosceles** is the closest classification, assuming approximate equality of two sides.
**Final answer:** Right, isosceles