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in δghi, h = 820 inches, m∠g = 102°, and m∠h = 10°. find the length of g, to the nearest inch.
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Explanation:

Step1: Recall the Law of Sines

The Law of Sines states that in any triangle, $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$, where $a, b, c$ are the lengths of the sides opposite angles $A, B, C$ respectively. In $\triangle GHI$, we have side $h = 820$ inches (opposite angle $H$), angle $G = 102^\circ$ (opposite side $g$), and angle $H = 10^\circ$ (opposite side $h$). So we can set up the proportion $\frac{g}{\sin G}=\frac{h}{\sin H}$.

Step2: Plug in the known values

We know $h = 820$, $\sin G=\sin(102^\circ)$, $\sin H=\sin(10^\circ)$. First, calculate the sines of the angles. $\sin(102^\circ)\approx\sin(78^\circ)\approx0.9781$ (since $\sin(180^\circ - x)=\sin x$), and $\sin(10^\circ)\approx0.1736$. Then, from the Law of Sines: $g=\frac{h\times\sin G}{\sin H}$. Substitute the values: $g=\frac{820\times0.9781}{0.1736}$.

Step3: Calculate the numerator and denominator

First, calculate the numerator: $820\times0.9781 = 820\times0.9781 = 802.042$. Then divide by the denominator: $\frac{802.042}{0.1736}\approx4620$ (rounded to the nearest inch). Wait, let's check the calculation again. Wait, $\sin(102^\circ)=\sin(180 - 78)= \sin78\approx0.9781$, $\sin(10^\circ)\approx0.1736$. So $820\times0.9781 = 820\times0.9781 = 820\times(1 - 0.0219)=820 - 820\times0.0219 = 820 - 17.958 = 802.042$. Then $802.042\div0.1736\approx4620$? Wait, no, wait, 0.1736 times 4620 is 0.1736*4620 = 0.1736*4000 + 0.1736*620 = 694.4 + 107.632 = 802.032, which matches the numerator. So $g\approx4620$ inches. Wait, but let's check the angle sum. The sum of angles in a triangle is $180^\circ$, so angle $I = 180 - 102 - 10 = 68^\circ$, but we don't need that for this calculation. Wait, maybe I made a mistake in the sine values? Let's use more precise values. $\sin(102^\circ)=\sin(102)=0.9781476007$, $\sin(10^\circ)=0.1736481777$. Then $g=\frac{820\times0.9781476007}{0.1736481777}$. Calculate numerator: $820\times0.9781476007 = 820\times0.9781476007 = 802.0810326$. Then divide by 0.1736481777: $802.0810326\div0.1736481777\approx4620$ (rounded to the nearest inch). Wait, but that seems very large. Wait, maybe I mixed up the sides. Wait, in triangle notation, side $a$ is opposite angle $A$, side $b$ opposite angle $B$, side $c$ opposite angle $C$. So in $\triangle GHI$, angle $G$ is at vertex $G$, so side opposite $G$ is $g$, side opposite $H$ is $h$, side opposite $I$ is $i$. So angle $H = 10^\circ$, side $h = 820$, angle $G = 102^\circ$, side $g$ is what we need. So Law of Sines: $\frac{g}{\sin G}=\frac{h}{\sin H}$. So $g = h\times\frac{\sin G}{\sin H}$. So $h = 820$, $\sin G=\sin102\approx0.9781$, $\sin H=\sin10\approx0.1736$. So $g = 820\times\frac{0.9781}{0.1736}\approx820\times5.633\approx820\times5.633 = 820\times5 + 820\times0.633 = 4100 + 519.06 = 4619.06\approx4619$ or $4620$ when rounded to the nearest inch. Wait, maybe my initial thought was wrong, but let's check with a calculator. Let's compute $\sin(102^\circ)= \sin(102)=0.9781476007$, $\sin(10^\circ)=0.1736481777$. Then $0.9781476007\div0.1736481777\approx5.633$. Then $820\times5.633 = 820\times5 + 820\times0.633 = 4100 + 519.06 = 4619.06\approx4619$ or $4620$? Wait, maybe I made a mistake in the angle labels. Wait, is side $h$ opposite angle $H$? Yes, in triangle notation, side $a$ is opposite angle $A$, so side $h$ is opposite angle $H$, side $g$ opposite angle $G$, side $i$ opposite angle $I$. So angle $H = 10^\circ$, side $h = 820$, angle $G = 102^\circ$, side $g$ is what we need. So the calculation is correct. So $g\approx4620$ inches? Wait, but that seems very large. Wait, maybe the angle $H$ is $10^\circ$, angle $G$ is $102^\circ$, so angle $I$ is $68^\circ$. So side opposite the largest angle (102°) is the longest side, which is $g$, and side $h$ is opposite 10°, which is the smallest angle, so it's the shortest side. So 820 is the shortest side, and $g$ is the longest side, so it should be longer than 820, which it is. So the calculation seems correct.

Answer:

$\boxed{4620}$ (Wait, but let's check with a calculator again. Let's use more precise values. $\sin(102^\circ)=0.9781476007$, $\sin(10^\circ)=0.1736481777$. Then $g = (820 * 0.9781476007)/0.1736481777 = (820 * 0.9781476007) / 0.1736481777$. Calculate 820 * 0.9781476007 = 802.0809326. Then divide by 0.1736481777: 802.0809326 / 0.1736481777 ≈ 4620. So the answer is approximately 4620 inches.