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In Triangle FGH, [tex]f = 910 \, \text{cm}[/tex], [tex]g = 190 \, \text{cm}[/tex], and [tex]h = 770 \, \text{cm}[/tex]. Find the measure of angle H to the nearest degree.

Answer :

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[tex]\textit{Law of Cosines}\\\\ \cfrac{a^2+b^2-c^2}{2ab}=\cos(C)\implies \cos^{-1}\left(\cfrac{a^2+b^2-c^2}{2ab}\right)=\measuredangle C \\\\[-0.35em] ~\dotfill\\\\ \cos^{-1}\left(\cfrac{190^2+910^2-770^2}{2(190)(910)}\right)=\measuredangle H \implies \cos^{-1}\left(\cfrac{ 271300 }{ 345800 }\right)=\measuredangle H \\\\\\ \cos^{-1}(0.7845575) \approx \measuredangle H \implies 38^o \approx \measuredangle H[/tex]

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Final answer:

Using the Law of Cosines, the measure of angle H in Triangle FGH is found to be approximately 125° to the nearest degree.

Explanation:

To find the measure of angle H in Triangle FGH with sides f = 910 cm, g = 190 cm, and h = 770 cm, we can use the Law of Cosines. This law relates the lengths of the sides of a triangle to the cosine of one of its angles. The formula for the Law of Cosines is: c² = a² + b² - 2ab*cos(C), where a, b, and c are the lengths of the sides of the triangle, and C is the angle opposite side c.

Applying the Law of Cosines to find angle H, we use the side lengths f (opposite angle F), g (opposite angle G), and h (opposite angle H). The formula becomes:

f² = g² + h² - 2*g*h*cos(H)

Rearranging the formula to solve for cos(H) gives us:

cos(H) = (g² + h² - f²)/(2*g*h)

Substitute the given side lengths:

cos(H) = (190² + 770² - 910²)/(2*190*770)

Calculate the value:

cos(H) = (36100 + 592900 - 828100)/(292600)

cos(H) = (-169100)/(292600)

cos(H) ≈ -0.5775

Now, we can find angle H by taking the arccos (inverse cosine) of cos(H):

H = arccos(-0.5775)

Calculating this value yields:

H ≈ 125° (to the nearest degree)

The measure of angle H in Triangle FGH to the nearest degree is approximately 125°.

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