In ΔFGH, f = 4.2 inches, g = 6.1 inches and ∠H=84°. Find the length of h, to the nearest 10th of an inch.

To find the length of side h in triangle FGH, we can use the Law of Cosines. The Law of Cosines states that for any triangle with sides of lengths a, b, and c, and an angle γ opposite side c, the following relationship holds:

c² = a² + b² - 2ab*cos(γ)

In this case, we know that a = f = 4.2 inches, b = g = 6.1 inches, and γ = ∠H = 84°. We want to find c = h. Plugging in the known values, we get:

h² = f² + g² - 2fgcos(H) h² = (4.2)² + (6.1)² - 2(4.2)*(6.1)*cos(84°)

First, calculate the squares: h² = 17.64 + 37.21 - 2*(4.2)*(6.1)*cos(84°)

Then, calculate the cosine: cos(84°) ≈ 0.104528463

So, plug this into the equation: h² = 17.64 + 37.21 - 2*(4.2)*(6.1)*0.104528463 h² = 54.85 - 5.28 h² = 49.57

Finally, take the square root of both sides to solve for h: h = √49.57 h ≈ 7.0 inches

So, the length of side h is approximately 7.0 inches.

To find the length of side h in triangle FGH, we can use the Law of Cosines. The Law of Cosines states that for any triangle with sides of lengths a, b, and c, and an angle γ opposite side c, the following equation holds true:

c² = a² + b² - 2ab*cos(γ)

In this case, we know the lengths of sides f and g, and the measure of angle H, so we can substitute these values into the equation:

h² = f² + g² - 2fg*cos(H)

Substituting the given values:

h² = (4.2)² + (6.1)² - 2*(4.2)*(6.1)*cos(84°)

First, calculate the squares:

h² = 17.64 + 37.21 - 2*(4.2)*(6.1)*cos(84°)

Then, calculate the cosine of 84 degrees. Make sure your calculator is in degree mode:

cos(84°) ≈ 0.104528

Substitute this value back into the equation:

h² = 17.64 + 37.21 - 2*(4.2)*(6.1)*0.104528

Simplify the equation:

h² = 54.85 - 5.22

Subtract to find the value of h²:

h² = 49.63

Finally, take the square root of both sides to solve for h:

h = √49.63 ≈ 7.0 inches

So, the length of side h is approximately 7.0 inches.