Question

In ΔFGH, f = 9.5 inches, g = 9 inches and h=9 inches. Find the measure of ∠H to the nearest degree.

Answer 1

To find the measure of \(\angle H\) in triangle \(\Delta FGH\) with side lengths \(f = 9.5\) inches, \(g = 9\) inches, and \(h = 9\) inches, we can use the Law of Cosines. According to this law, the relationship between the sides and the angles of a triangle is given by the formula:

\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]

For triangle \(\Delta FGH\), we will use the following assignment for the sides:
- \(\angle H\) is opposite side \(h\),
- \(\angle F\) is opposite side \(f\),
- \(\angle G\) is opposite side \(g\).

Applying the Law of Cosines, we have:

\[
h^2 = f^2 + g^2 - 2fg \cdot \cos(H)
\]

Substituting the known values:

\[
9^2 = 9.5^2 + 9^2 - 2 \cdot 9.5 \cdot 9 \cdot \cos(H)
\]

Calculating the squares:

\[
81 = 90.25 + 81 - 2 \cdot 9.5 \cdot 9 \cdot \cos(H)
\]

Adding \(90.25 + 81\):

\[
81 = 171.25 - 2 \cdot 9.5 \cdot 9 \cdot \cos(H)
\]

Now, rearranging the equation:

\[
2 \cdot 9.5 \cdot 9 \cdot \cos(H) = 171.25 - 81
\]

This simplifies to:

\[
2 \cdot 9.5 \cdot 9 \cdot \cos(H) = 90.25
\]

Calculating \(2 \cdot 9.5 \cdot 9\):

\[
2 \cdot 9.5 \cdot 9 = 171
\]

So, the equation becomes:

\[
171 \cdot \cos(H) = 90.25
\]

Now we solve for \(\cos(H)\):

\[
\cos(H) = \frac{90.25}{171}
\]

Calculating that value:

\[
\cos(H) \approx 0.5275
\]

Now, we will find \(\angle H\):

\[
H = \cos^{-1}(0.5275)
\]

Using a calculator to find \(\angle H\):

\[
H \approx 58.5^\circ
\]

Rounding to the nearest degree, we find:

\[
\angle H \approx 59^\circ
\]

Thus, the measure of \(\angle H\) to the nearest degree is \(\boxed{59}\).